Notifications
Retirer tout
Forum Roulette
99
Posts
9
Utilisateurs
0
Reactions
16.9 {numéro}K
Vu
17/12/2012 9:14 am
Ton calcul de l'espérance mathématique est faux.
Décidément, c'est une obsession !
Déjà expliqué dans au moins 2 messages auparavant mais .... tu es long à la détente.
Le bilan est donc négatif et on perd en moyenne -0.27027027 mise dans un cycle de 20 spins.
Autrement dit, on perd -1.351% environ.
-0.27027027 est l'espérance pour 20 lancers
Autrement dit, l'espérance (par lancer) est -1/74
17/12/2012 9:41 am
Tu m'as posé la question concernant la loi que tu utilises.Il s'agit de la loi multinomiale.
Donc j'ai refait les calculs selon la loi multinomiale et voici ce que je trouve :
Probabilité du gain : +0.438101948114484940
Probabilité de la perte : +0.459744483904462600
Oui, tu viens de refaire exactement mes calculs.
Mais tu ne dis pas finalement lesquels sont les bons, les tiens ou les miens ?
17/12/2012 8:28 pm
Après avoir refait tes calculs, je constate qu'il y a quelque chose qui me dérange !
Normalement l'avantage du zéro se fait sur un seul coup et non sur tous les coups d'une série donnée.
Or dans le calcul, nous avons une série de vingt coups et nous calculons une probabilité unitaire, mais globale à cette série.
L'espérance mathématique confirme (???) que ce résultat est correcte.
Donc, je considère que l'influence du zéro se répartie sur la totalité de ces vingt coups, sans changer le résultat.
De cela, le résultat est juste à la condition de considérer qu'il s'agisse d'un jeu à masse égale.
Et que l'ordre d'apparition des coups n'a donc pas d'importance puisque les mises sont constantes !
L'autre point que je soulève encore est l'importance de la valeur donnée à la probabilité du gain.
Pour comparer des résultats, il faut les comparer sur les mêmes bases.
Dans mon calcul binomial, j'ai comparé les trois cas suivants et j'ai trouvé la relation suivante :
Roulette européenne < Roulette française < jeu à Pile ou Face.
De ce fait, mon calcul binomiale de la roulette française est cohérent !
Maintenant si tu compares ton résultat avec ceux issues du calcul binomial, cela n'est plus cohérent.
Or nous jouons sur une roulette !
Donc il faut prendre les trois probabilités suivantes :
--> Prob[Zéro] = 1 / 37.
--> Prob[Rouge] = 18 / 37.
--> Prob[Noir] = 18 / 37.
et appliquer la loi multinomiale.
Et gérer les mises comme si nous avions :
--> une roulette sans zéro.
--> une roulette avec un zéro sans avantage (roulette européenne).
--> une roulette avec un zéro avec avantage (roulette française).
Car faire un calcul avec la loi binomiale et la comparer avec la loi multinomiale, cela n'a pas de sens.
Oui, j'ai fait précédemment cette comparaison. Et je constate qu'elle est fausse.
Donc si le seul critère est l'espérance mathématique, nous pouvons confirmer que la roulette française est le meilleur jeu.
Pourquoi ? Car nous avons la même base de calcul.
Pour la probabilité des gains, je suis d'accord pour ceux relatifs au jeu de Pile ou Face, et de la roulette européenne.
Mais je ne suis pas du même avis pour la roulette française.
Donc pour répondre à ta question, le calcul de la loi multinomiale semble plus juste.
Mais un calcul seul ne signifie rien. Pour le mesurer, il faut le comparer ! Donc avoir la même base de comparaison.
Or c'est justement le problème de shynx, la comparaison !
@+
17/12/2012 8:42 pm
Je n'ai quasiment rien compris à ton long discours de 32 lignes.
Je ne retiendrai que la trentième:
Donc pour répondre à ta question, le calcul de la loi multinomiale semble plus juste.
Tu valides donc que pour une française, il y a 43.81% de cycles de 20 lancers gagnants.
Or tu trouves 0,4381. Ce résultat est meilleur que le jeu de Pile ou face et de ce fait, il est aberrant car le jeu de la roulette française à une espérance mathématique négative alors que Pile ou face a une espérance mathématique nulle.
Tu ne trouves plus ça aberrant ?
17/12/2012 8:47 pm
En quelle langue faut il que je te parle ?
Si tu prenais le temps de comprendre, on n'aurait pas tous ces quiproquo ou ces contresens !
Je dis qu'il faut une base de comparaison !
Si tu donnes un résultat seul, cela ne veut strictement rien dire.
17/12/2012 8:50 pm
En quelle langue faut il que je te parle ?
Si tu prenais le temps de comprendre, on n'aurait pas tous ces quiproquo ou ces contresens !.
Je viens de relire
Ce que tu écris est incompréhensible..
Je dis qu'il faut une base de comparaison !
Si tu donnes un résultat seul, cela ne veut strictement rien dire.
Une base de comparaison ?
Tu veux comparer la roulette française avec la pièce?
Je l'ai déjà fait et cela ne pose absolument aucun problème.
Pour une française :
Il y a 43.81% de cycles gagnants pendant lesquels on gagne en moyenne 1.62686211 mise
Il y a 45.97% de cycles perdants pendant lesquels on perd en moyenne -1.89713238 mise
Bilan: l'espérance de gain est bien -1/74 par lancer.
Pour un jeu de pile ou face :
Il y a 41.19% de cycles gagnants pendant lesquels on gagne en moyenne 1.76197052 mise
Il y a 41.19% de cycles perdants pendant lesquels on perd en moyenne -1.76197052mise
Bilan: l'espérance de gain est bien 0
Où est le problème ? Il n'y a aucun problème !
18/12/2012 8:16 pm
Je dis qu'il faut une base de comparaison !
Si tu donnes un résultat seul, cela ne veut strictement rien dire.Une base de comparaison ?
Tu veux comparer la roulette française avec la pièce?
Tu le fais exprès ou quoi ? Qui te parle de la comparaison avec une pièce ?
Comparer la roulette française avec la roulette européenne via la loi multinomiale sur les trois probabilités suivantes ;
--> Prob[Zéro] = 1 / 37.
--> Prob[Rouge] = 18 / 37.
--> Prob[Noir] = 18 / 37.
c'est de cela dont je parle. J'ai l'impression que tu ne comprends pas le française, mon cher Glups.
La différence dans les deux calculs, seul le "bilan" des jetons change.
Dans les deux cas, tu vas trouver une probabilité des gains et tu es alors en mesure de dire laquelle des deux roulettes est la plus performante.
Mais donner une probabilité des gains sur la loi binomiale et ensuite donner une probabilité des gains sur la loi multinomiale et indiquer que la seconde est meilleure que la première n'a pas de sens. Il faut la même base de calcul.
C'est comme si tu disais que 3 kilomètres, c'est plus petit que 5 kilogrammes ! Ça y est, as-tu enfin compris ?
Encore une fois, je ne critique pas tes calculs, mais le sens que tu leur attributs ! C'est ce manque de rigueur qui m'énerve.
@+
18/12/2012 8:35 pm
Tu le fais exprès ou quoi ? Qui te parle de la comparaison avec une pièce ?
C'est toi qui a parlé de pièce.
On se trouve alors en présence de 3 jeux.
Tu parles de comparaison sans préciser ce que tu compares (voir plus bas).
Comparer la roulette française avec la roulette européenne via la loi multinomiale sur les trois probabilités suivantes ;
Je pouvais considérer que tu avais un problème entre la française et la pièce car tu disais qu'il était aberrant de trouver plus de cycles gagnants avec la française qu' avec la pièce. Reconnais que ce n'est pas clair !
Mais entre l' européenne et la française, il y a encore moins de problème de comparaison.
Mais donner une probabilité des gains sur la loi binomiale et ensuite donner une probabilité des gains sur la loi multinomiale et indiquer que la seconde est meilleure que la première n'a pas de sens. Il faut la même base de calcul.
Ok, ce qui te gêne cette fois, c'est qu'on n'utilise pas la même loi pour calculer les probabilités ? C'est ça ?
Mais donner une probabilité des gains sur la loi binomiale et ensuite donner une probabilité des gains sur la loi multinomiale et indiquer que la seconde est meilleure que la première n'a pas de sens. Il faut la même base de calcul.
On ne calcule pas la probabilité sur une loi.
C'est une loi qui permet de calculer la probabilité. Mais pour calculer la probabilité, il faut prendre la loi appropriée.
La roulette européenne a 2 retours, on utilise la loi binomiale.
La roulette française a 3 retours, on utilise la loi multinomiale.
Il faut prendre l'outil adapté sinon le résultat ne sera pas bon.
Et pour la roulette française, il n'y a pas 2 résultats.
19/12/2012 2:53 am
On ne calcule pas la probabilité sur une loi.
C'est une loi qui permet de calculer la probabilité. Mais pour calculer la probabilité, il faut prendre la loi appropriée.
Qu'est-ce que c'est que ce charabia que tu nous racontes ?
La roulette européenne a 2 retours, on utilise la loi binomiale.
La roulette française a 3 retours, on utilise la loi multinomiale.
Il faut prendre l'outil adapté sinon tes résultats ne seront pas bons.
Le problème, ce n'est pas le nombre de retours comme tu sembles l'indiquer.
Le problème, ce sont les probabilités.
Pour la roulette française et la roulette européenne, il y a trois probabilité :
--> Prob[Zéro] = 1 / 37.
--> Prob[Rouge] = 18 / 37.
--> Prob[Noir] = 18 / 37.
Donc on peut utiliser la loi multinomiale. As-tu compris ?
Et ainsi on peut comparer la roulette européenne avec la roulette française !
On peut aussi simplifier la roulette européenne, en ne prenant que deux probabilités :
--> Prob[Rouge] = 18 / 37.
--> Prob[Noir] = 19 / 37.
à la condition de miser sur Rouge, bien sûr.
Et tu peux comparer avec le jeu de Pile ou Face (je suppose que c'est ce que tu appelle Pièce) :
--> Prob[Rouge] = 1 / 2.
--> Prob[Noir] = 1 / 2.
Et ici, tu utilises la loi binomiale. As-tu compris ?
Mais tu ne peux pas comparer la loi binomiale sur le jeu de Pile ou Face avec la loi multinomiale avec la roulette française.
Pourquoi ? Car ce n'est pas la même loi, et ce ne sont pas les mêmes probabilités.
Donc si tu veux comparer le jeu de Pile ou face avec la roulette française, tu dois prendre les mêmes probabilités :
--> Prob[Zéro] = 1 / 37.
--> Prob[Rouge] = 18 / 37.
--> Prob[Noir] = 18 / 37.
et faire en sorte, que le bilan soit adapter à ces deux jeux.
Par exemple, au jeu de pile ou face, si le zéro sort, le bilan est zéro !
C'est bon maintenant !
Et voici le calcul, avec la loi multinomiale, selon le jeu de Pile ou face.
Triplets | Combinaisons | Probabilités | Bilan | Espérances Mathématiques | ============== | ============ | ================================================== | ===== | =================================================== | ( 0 ; 0 ; 20) | 1 | 0.000000551335072831166730000000000000000000000000 | -20.0 | -0.000000551335072831166730000000000000000000000000 | ( 0 ; 1 ; 19) | 20 | 0.000011026701456623334000000000000000000000000000 | -18.0 | -0.000009924031310961000900000000000000000000000000 | ( 0 ; 2 ; 18) | 190 | 0.000104753663837921680000000000000000000000000000 | -16.0 | -0.000083802931070337347000000000000000000000000000 | ( 0 ; 3 ; 17) | 1140 | 0.000628521983027530090000000000000000000000000000 | -14.0 | -0.000439965388119271050000000000000000000000000000 | ( 0 ; 4 ; 16) | 4845 | 0.002671218427867002600000000000000000000000000000 | -12.0 | -0.001602731056720201500000000000000000000000000000 | ( 0 ; 5 ; 15) | 15504 | 0.008547898969174409100000000000000000000000000000 | -10.0 | -0.004273949484587204500000000000000000000000000000 | ( 0 ; 6 ; 14) | 38760 | 0.021369747422936021000000000000000000000000000000 | -8.0 | -0.008547898969174409100000000000000000000000000000 | ( 0 ; 7 ; 13) | 77520 | 0.042739494845872042000000000000000000000000000000 | -6.0 | -0.012821848453761612000000000000000000000000000000 | ( 0 ; 8 ; 12) | 125970 | 0.069451679124542073000000000000000000000000000000 | -4.0 | -0.013890335824908414000000000000000000000000000000 | ( 0 ; 9 ; 11) | 167960 | 0.092602238832722755000000000000000000000000000000 | -2.0 | -0.009260223883272276200000000000000000000000000000 | ( 0 ; 10 ; 10) | 184756 | 0.101862462715995030000000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 0 ; 11 ; 9) | 167960 | 0.092602238832722755000000000000000000000000000000 | +2.0 | +0.009260223883272276200000000000000000000000000000 | ( 0 ; 12 ; 8) | 125970 | 0.069451679124542073000000000000000000000000000000 | +4.0 | +0.013890335824908414000000000000000000000000000000 | ( 0 ; 13 ; 7) | 77520 | 0.042739494845872042000000000000000000000000000000 | +6.0 | +0.012821848453761612000000000000000000000000000000 | ( 0 ; 14 ; 6) | 38760 | 0.021369747422936021000000000000000000000000000000 | +8.0 | +0.008547898969174409100000000000000000000000000000 | ( 0 ; 15 ; 5) | 15504 | 0.008547898969174409100000000000000000000000000000 | +10.0 | +0.004273949484587204500000000000000000000000000000 | ( 0 ; 16 ; 4) | 4845 | 0.002671218427867002600000000000000000000000000000 | +12.0 | +0.001602731056720201500000000000000000000000000000 | ( 0 ; 17 ; 3) | 1140 | 0.000628521983027530090000000000000000000000000000 | +14.0 | +0.000439965388119271050000000000000000000000000000 | ( 0 ; 18 ; 2) | 190 | 0.000104753663837921680000000000000000000000000000 | +16.0 | +0.000083802931070337347000000000000000000000000000 | ( 0 ; 19 ; 1) | 20 | 0.000011026701456623334000000000000000000000000000 | +18.0 | +0.000009924031310961000900000000000000000000000000 | ( 0 ; 20 ; 0) | 1 | 0.000000551335072831166730000000000000000000000000 | +20.0 | +0.000000551335072831166730000000000000000000000000 | ( 1 ; 0 ; 19) | 20 | 0.000000612594525367962980000000000000000000000000 | -19.0 | -0.000000581964799099564800000000000000000000000000 | ( 1 ; 1 ; 18) | 380 | 0.000011639295981991295000000000000000000000000000 | -17.0 | -0.000009893401584692600000000000000000000000000000 | ( 1 ; 2 ; 17) | 3420 | 0.000104753663837921660000000000000000000000000000 | -15.0 | -0.000078565247878441247000000000000000000000000000 | ( 1 ; 3 ; 16) | 19380 | 0.000593604095081556200000000000000000000000000000 | -13.0 | -0.000385842661803011550000000000000000000000000000 | ( 1 ; 4 ; 15) | 77520 | 0.002374416380326224800000000000000000000000000000 | -11.0 | -0.001305929009179423700000000000000000000000000000 | ( 1 ; 5 ; 14) | 232560 | 0.007123249140978673900000000000000000000000000000 | -9.0 | -0.003205462113440403400000000000000000000000000000 | ( 1 ; 6 ; 13) | 542640 | 0.016620914662283572000000000000000000000000000000 | -7.0 | -0.005817320131799250300000000000000000000000000000 | ( 1 ; 7 ; 12) | 1007760 | 0.030867412944240915000000000000000000000000000000 | -5.0 | -0.007716853236060228700000000000000000000000000000 | ( 1 ; 8 ; 11) | 1511640 | 0.046301119416361385000000000000000000000000000000 | -3.0 | -0.006945167912454208000000000000000000000000000000 | ( 1 ; 9 ; 10) | 1847560 | 0.056590257064441686000000000000000000000000000000 | -1.0 | -0.002829512853222084300000000000000000000000000000 | ( 1 ; 10 ; 9) | 1847560 | 0.056590257064441686000000000000000000000000000000 | +1.0 | +0.002829512853222084300000000000000000000000000000 | ( 1 ; 11 ; 8) | 1511640 | 0.046301119416361385000000000000000000000000000000 | +3.0 | +0.006945167912454208000000000000000000000000000000 | ( 1 ; 12 ; 7) | 1007760 | 0.030867412944240918000000000000000000000000000000 | +5.0 | +0.007716853236060229600000000000000000000000000000 | ( 1 ; 13 ; 6) | 542640 | 0.016620914662283572000000000000000000000000000000 | +7.0 | +0.005817320131799250300000000000000000000000000000 | ( 1 ; 14 ; 5) | 232560 | 0.007123249140978673900000000000000000000000000000 | +9.0 | +0.003205462113440403400000000000000000000000000000 | ( 1 ; 15 ; 4) | 77520 | 0.002374416380326224800000000000000000000000000000 | +11.0 | +0.001305929009179423700000000000000000000000000000 | ( 1 ; 16 ; 3) | 19380 | 0.000593604095081556200000000000000000000000000000 | +13.0 | +0.000385842661803011550000000000000000000000000000 | ( 1 ; 17 ; 2) | 3420 | 0.000104753663837921660000000000000000000000000000 | +15.0 | +0.000078565247878441247000000000000000000000000000 | ( 1 ; 18 ; 1) | 380 | 0.000011639295981991296000000000000000000000000000 | +17.0 | +0.000009893401584692601700000000000000000000000000 | ( 1 ; 19 ; 0) | 20 | 0.000000612594525367962980000000000000000000000000 | +19.0 | +0.000000581964799099564800000000000000000000000000 | ( 2 ; 0 ; 18) | 190 | 0.000000323313777277536050000000000000000000000000 | -18.0 | -0.000000290982399549782450000000000000000000000000 | ( 2 ; 1 ; 17) | 3420 | 0.000005819647990995648200000000000000000000000000 | -16.0 | -0.000004655718392796518400000000000000000000000000 | ( 2 ; 2 ; 16) | 29070 | 0.000049467007923463012000000000000000000000000000 | -14.0 | -0.000034626905546424111000000000000000000000000000 | ( 2 ; 3 ; 15) | 155040 | 0.000263824042258469420000000000000000000000000000 | -12.0 | -0.000158294425355081650000000000000000000000000000 | ( 2 ; 4 ; 14) | 581400 | 0.000989340158469260290000000000000000000000000000 | -10.0 | -0.000494670079234630150000000000000000000000000000 | ( 2 ; 5 ; 13) | 1627920 | 0.002770152443713928600000000000000000000000000000 | -8.0 | -0.001108060977485571300000000000000000000000000000 | ( 2 ; 6 ; 12) | 3527160 | 0.006001996961380179400000000000000000000000000000 | -6.0 | -0.001800599088414053800000000000000000000000000000 | ( 2 ; 7 ; 11) | 6046560 | 0.010289137648080307000000000000000000000000000000 | -4.0 | -0.002057827529616061400000000000000000000000000000 | ( 2 ; 8 ; 10) | 8314020 | 0.014147564266110422000000000000000000000000000000 | -2.0 | -0.001414756426611042200000000000000000000000000000 | ( 2 ; 9 ; 9) | 9237800 | 0.015719515851233802000000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 2 ; 10 ; 8) | 8314020 | 0.014147564266110422000000000000000000000000000000 | +2.0 | +0.001414756426611042200000000000000000000000000000 | ( 2 ; 11 ; 7) | 6046560 | 0.010289137648080307000000000000000000000000000000 | +4.0 | +0.002057827529616061400000000000000000000000000000 | ( 2 ; 12 ; 6) | 3527160 | 0.006001996961380179400000000000000000000000000000 | +6.0 | +0.001800599088414053800000000000000000000000000000 | ( 2 ; 13 ; 5) | 1627920 | 0.002770152443713928600000000000000000000000000000 | +8.0 | +0.001108060977485571300000000000000000000000000000 | ( 2 ; 14 ; 4) | 581400 | 0.000989340158469260290000000000000000000000000000 | +10.0 | +0.000494670079234630150000000000000000000000000000 | ( 2 ; 15 ; 3) | 155040 | 0.000263824042258469420000000000000000000000000000 | +12.0 | +0.000158294425355081650000000000000000000000000000 | ( 2 ; 16 ; 2) | 29070 | 0.000049467007923463019000000000000000000000000000 | +14.0 | +0.000034626905546424111000000000000000000000000000 | ( 2 ; 17 ; 1) | 3420 | 0.000005819647990995648200000000000000000000000000 | +16.0 | +0.000004655718392796518400000000000000000000000000 | ( 2 ; 18 ; 0) | 190 | 0.000000323313777277536050000000000000000000000000 | +18.0 | +0.000000290982399549782450000000000000000000000000 | ( 3 ; 0 ; 17) | 1140 | 0.000000107771259092512000000000000000000000000000 | -17.0 | -0.000000091605570228635205000000000000000000000000 | ( 3 ; 1 ; 16) | 19380 | 0.000001832111404572704000000000000000000000000000 | -15.0 | -0.000001374083553429527900000000000000000000000000 | ( 3 ; 2 ; 15) | 155040 | 0.000014656891236581632000000000000000000000000000 | -13.0 | -0.000009526979303778061200000000000000000000000000 | ( 3 ; 3 ; 14) | 775200 | 0.000073284456182908165000000000000000000000000000 | -11.0 | -0.000040306450900599488000000000000000000000000000 | ( 3 ; 4 ; 13) | 2713200 | 0.000256495596640178590000000000000000000000000000 | -9.0 | -0.000115423018488080370000000000000000000000000000 | ( 3 ; 5 ; 12) | 7054320 | 0.000666888551264464230000000000000000000000000000 | -7.0 | -0.000233410992942562490000000000000000000000000000 | ( 3 ; 6 ; 11) | 14108640 | 0.001333777102528928500000000000000000000000000000 | -5.0 | -0.000333444275632232110000000000000000000000000000 | ( 3 ; 7 ; 10) | 22170720 | 0.002095935446831173300000000000000000000000000000 | -3.0 | -0.000314390317024675990000000000000000000000000000 | ( 3 ; 8 ; 9) | 27713400 | 0.002619919308538966600000000000000000000000000000 | -1.0 | -0.000130995965426948330000000000000000000000000000 | ( 3 ; 9 ; 8) | 27713400 | 0.002619919308538966600000000000000000000000000000 | +1.0 | +0.000130995965426948330000000000000000000000000000 | ( 3 ; 10 ; 7) | 22170720 | 0.002095935446831173300000000000000000000000000000 | +3.0 | +0.000314390317024675990000000000000000000000000000 | ( 3 ; 11 ; 6) | 14108640 | 0.001333777102528928500000000000000000000000000000 | +5.0 | +0.000333444275632232110000000000000000000000000000 | ( 3 ; 12 ; 5) | 7054320 | 0.000666888551264464230000000000000000000000000000 | +7.0 | +0.000233410992942562490000000000000000000000000000 | ( 3 ; 13 ; 4) | 2713200 | 0.000256495596640178590000000000000000000000000000 | +9.0 | +0.000115423018488080370000000000000000000000000000 | ( 3 ; 14 ; 3) | 775200 | 0.000073284456182908165000000000000000000000000000 | +11.0 | +0.000040306450900599488000000000000000000000000000 | ( 3 ; 15 ; 2) | 155040 | 0.000014656891236581631000000000000000000000000000 | +13.0 | +0.000009526979303778059500000000000000000000000000 | ( 3 ; 16 ; 1) | 19380 | 0.000001832111404572704000000000000000000000000000 | +15.0 | +0.000001374083553429527900000000000000000000000000 | ( 3 ; 17 ; 0) | 1140 | 0.000000107771259092512000000000000000000000000000 | +17.0 | +0.000000091605570228635205000000000000000000000000 | ( 4 ; 0 ; 16) | 4845 | 0.000000025445991730176449000000000000000000000000 | -16.0 | -0.000000020356793384141160000000000000000000000000 | ( 4 ; 1 ; 15) | 77520 | 0.000000407135867682823190000000000000000000000000 | -14.0 | -0.000000284995107377976230000000000000000000000000 | ( 4 ; 2 ; 14) | 581400 | 0.000003053519007621173400000000000000000000000000 | -12.0 | -0.000001832111404572704000000000000000000000000000 | ( 4 ; 3 ; 13) | 2713200 | 0.000014249755368898811000000000000000000000000000 | -10.0 | -0.000007124877684449405300000000000000000000000000 | ( 4 ; 4 ; 12) | 8817900 | 0.000046311704948921134000000000000000000000000000 | -8.0 | -0.000018524681979568454000000000000000000000000000 | ( 4 ; 5 ; 11) | 21162960 | 0.000111148091877410710000000000000000000000000000 | -6.0 | -0.000033344427563223214000000000000000000000000000 | ( 4 ; 6 ; 10) | 38798760 | 0.000203771501775252960000000000000000000000000000 | -4.0 | -0.000040754300355050594000000000000000000000000000 | ( 4 ; 7 ; 9) | 55426800 | 0.000291102145393218570000000000000000000000000000 | -2.0 | -0.000029110214539321859000000000000000000000000000 | ( 4 ; 8 ; 8) | 62355150 | 0.000327489913567370880000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 4 ; 9 ; 7) | 55426800 | 0.000291102145393218520000000000000000000000000000 | +2.0 | +0.000029110214539321852000000000000000000000000000 | ( 4 ; 10 ; 6) | 38798760 | 0.000203771501775252960000000000000000000000000000 | +4.0 | +0.000040754300355050594000000000000000000000000000 | ( 4 ; 11 ; 5) | 21162960 | 0.000111148091877410720000000000000000000000000000 | +6.0 | +0.000033344427563223214000000000000000000000000000 | ( 4 ; 12 ; 4) | 8817900 | 0.000046311704948921134000000000000000000000000000 | +8.0 | +0.000018524681979568454000000000000000000000000000 | ( 4 ; 13 ; 3) | 2713200 | 0.000014249755368898811000000000000000000000000000 | +10.0 | +0.000007124877684449405300000000000000000000000000 | ( 4 ; 14 ; 2) | 581400 | 0.000003053519007621173400000000000000000000000000 | +12.0 | +0.000001832111404572704000000000000000000000000000 | ( 4 ; 15 ; 1) | 77520 | 0.000000407135867682823140000000000000000000000000 | +14.0 | +0.000000284995107377976180000000000000000000000000 | ( 4 ; 16 ; 0) | 4845 | 0.000000025445991730176449000000000000000000000000 | +16.0 | +0.000000020356793384141160000000000000000000000000 | ( 5 ; 0 ; 15) | 15504 | 0.000000004523731863142479100000000000000000000000 | -15.0 | -0.000000003392798897356859100000000000000000000000 | ( 5 ; 1 ; 14) | 232560 | 0.000000067855977947137198000000000000000000000000 | -13.0 | -0.000000044106385665639179000000000000000000000000 | ( 5 ; 2 ; 13) | 1627920 | 0.000000474991845629960390000000000000000000000000 | -11.0 | -0.000000261245515096478190000000000000000000000000 | ( 5 ; 3 ; 12) | 7054320 | 0.000002058297997729827800000000000000000000000000 | -9.0 | -0.000000926234098978422520000000000000000000000000 | ( 5 ; 4 ; 11) | 21162960 | 0.000006174893993189484300000000000000000000000000 | -7.0 | -0.000002161212897616319600000000000000000000000000 | ( 5 ; 5 ; 10) | 46558512 | 0.000013584766785016867000000000000000000000000000 | -5.0 | -0.000003396191696254216800000000000000000000000000 | ( 5 ; 6 ; 9) | 77597520 | 0.000022641277975028109000000000000000000000000000 | -3.0 | -0.000003396191696254216300000000000000000000000000 | ( 5 ; 7 ; 8) | 99768240 | 0.000029110214539321855000000000000000000000000000 | -1.0 | -0.000001455510726966092800000000000000000000000000 | ( 5 ; 8 ; 7) | 99768240 | 0.000029110214539321852000000000000000000000000000 | +1.0 | +0.000001455510726966092600000000000000000000000000 | ( 5 ; 9 ; 6) | 77597520 | 0.000022641277975028109000000000000000000000000000 | +3.0 | +0.000003396191696254216300000000000000000000000000 | ( 5 ; 10 ; 5) | 46558512 | 0.000013584766785016867000000000000000000000000000 | +5.0 | +0.000003396191696254216800000000000000000000000000 | ( 5 ; 11 ; 4) | 21162960 | 0.000006174893993189484300000000000000000000000000 | +7.0 | +0.000002161212897616319600000000000000000000000000 | ( 5 ; 12 ; 3) | 7054320 | 0.000002058297997729827800000000000000000000000000 | +9.0 | +0.000000926234098978422520000000000000000000000000 | ( 5 ; 13 ; 2) | 1627920 | 0.000000474991845629960340000000000000000000000000 | +11.0 | +0.000000261245515096478190000000000000000000000000 | ( 5 ; 14 ; 1) | 232560 | 0.000000067855977947137198000000000000000000000000 | +13.0 | +0.000000044106385665639179000000000000000000000000 | ( 5 ; 15 ; 0) | 15504 | 0.000000004523731863142479100000000000000000000000 | +15.0 | +0.000000003392798897356859100000000000000000000000 | ( 6 ; 0 ; 14) | 38760 | 0.000000000628296092103122160000000000000000000000 | -14.0 | -0.000000000439807264472185530000000000000000000000 | ( 6 ; 1 ; 13) | 542640 | 0.000000008796145289443711100000000000000000000000 | -12.0 | -0.000000005277687173666226800000000000000000000000 | ( 6 ; 2 ; 12) | 3527160 | 0.000000057174944381384111000000000000000000000000 | -10.0 | -0.000000028587472190692056000000000000000000000000 | ( 6 ; 3 ; 11) | 14108640 | 0.000000228699777525536420000000000000000000000000 | -8.0 | -0.000000091479911010214573000000000000000000000000 | ( 6 ; 4 ; 10) | 38798760 | 0.000000628924388195225300000000000000000000000000 | -6.0 | -0.000000188677316458567590000000000000000000000000 | ( 6 ; 5 ; 9) | 77597520 | 0.000001257848776390450600000000000000000000000000 | -4.0 | -0.000000251569755278090110000000000000000000000000 | ( 6 ; 6 ; 8) | 116396280 | 0.000001886773164585675600000000000000000000000000 | -2.0 | -0.000000188677316458567570000000000000000000000000 | ( 6 ; 7 ; 7) | 133024320 | 0.000002156312188097915100000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 6 ; 8 ; 6) | 116396280 | 0.000001886773164585675600000000000000000000000000 | +2.0 | +0.000000188677316458567570000000000000000000000000 | ( 6 ; 9 ; 5) | 77597520 | 0.000001257848776390450600000000000000000000000000 | +4.0 | +0.000000251569755278090110000000000000000000000000 | ( 6 ; 10 ; 4) | 38798760 | 0.000000628924388195225300000000000000000000000000 | +6.0 | +0.000000188677316458567590000000000000000000000000 | ( 6 ; 11 ; 3) | 14108640 | 0.000000228699777525536450000000000000000000000000 | +8.0 | +0.000000091479911010214573000000000000000000000000 | ( 6 ; 12 ; 2) | 3527160 | 0.000000057174944381384105000000000000000000000000 | +10.0 | +0.000000028587472190692052000000000000000000000000 | ( 6 ; 13 ; 1) | 542640 | 0.000000008796145289443709400000000000000000000000 | +12.0 | +0.000000005277687173666226000000000000000000000000 | ( 6 ; 14 ; 0) | 38760 | 0.000000000628296092103122160000000000000000000000 | +14.0 | +0.000000000439807264472185530000000000000000000000 | ( 7 ; 0 ; 13) | 77520 | 0.000000000069810676900346914000000000000000000000 | -13.0 | -0.000000000045376939985225495000000000000000000000 | ( 7 ; 1 ; 12) | 1007760 | 0.000000000907538799704509870000000000000000000000 | -11.0 | -0.000000000499146339837480420000000000000000000000 | ( 7 ; 2 ; 11) | 6046560 | 0.000000005445232798227058200000000000000000000000 | -9.0 | -0.000000002450354759202176100000000000000000000000 | ( 7 ; 3 ; 10) | 22170720 | 0.000000019965853593499215000000000000000000000000 | -7.0 | -0.000000006988048757724725500000000000000000000000 | ( 7 ; 4 ; 9) | 55426800 | 0.000000049914633983748041000000000000000000000000 | -5.0 | -0.000000012478658495937010000000000000000000000000 | ( 7 ; 5 ; 8) | 99768240 | 0.000000089846341170746465000000000000000000000000 | -3.0 | -0.000000013476951175611969000000000000000000000000 | ( 7 ; 6 ; 7) | 133024320 | 0.000000119795121560995300000000000000000000000000 | -1.0 | -0.000000005989756078049765100000000000000000000000 | ( 7 ; 7 ; 6) | 133024320 | 0.000000119795121560995280000000000000000000000000 | +1.0 | +0.000000005989756078049764200000000000000000000000 | ( 7 ; 8 ; 5) | 99768240 | 0.000000089846341170746465000000000000000000000000 | +3.0 | +0.000000013476951175611969000000000000000000000000 | ( 7 ; 9 ; 4) | 55426800 | 0.000000049914633983748041000000000000000000000000 | +5.0 | +0.000000012478658495937010000000000000000000000000 | ( 7 ; 10 ; 3) | 22170720 | 0.000000019965853593499215000000000000000000000000 | +7.0 | +0.000000006988048757724725500000000000000000000000 | ( 7 ; 11 ; 2) | 6046560 | 0.000000005445232798227058200000000000000000000000 | +9.0 | +0.000000002450354759202176100000000000000000000000 | ( 7 ; 12 ; 1) | 1007760 | 0.000000000907538799704509870000000000000000000000 | +11.0 | +0.000000000499146339837480420000000000000000000000 | ( 7 ; 13 ; 0) | 77520 | 0.000000000069810676900346914000000000000000000000 | +13.0 | +0.000000000045376939985225495000000000000000000000 | ( 8 ; 0 ; 12) | 125970 | 0.000000000006302352775725761800000000000000000000 | -12.0 | -0.000000000003781411665435457100000000000000000000 | ( 8 ; 1 ; 11) | 1511640 | 0.000000000075628233308709139000000000000000000000 | -10.0 | -0.000000000037814116654354569000000000000000000000 | ( 8 ; 2 ; 10) | 8314020 | 0.000000000415955283197900350000000000000000000000 | -8.0 | -0.000000000166382113279160140000000000000000000000 | ( 8 ; 3 ; 9) | 27713400 | 0.000000001386517610659667600000000000000000000000 | -6.0 | -0.000000000415955283197900300000000000000000000000 | ( 8 ; 4 ; 8) | 62355150 | 0.000000003119664623984252200000000000000000000000 | -4.0 | -0.000000000623932924796850480000000000000000000000 | ( 8 ; 5 ; 7) | 99768240 | 0.000000004991463398374803000000000000000000000000 | -2.0 | -0.000000000499146339837480320000000000000000000000 | ( 8 ; 6 ; 6) | 116396280 | 0.000000005823373964770603900000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 8 ; 7 ; 5) | 99768240 | 0.000000004991463398374803000000000000000000000000 | +2.0 | +0.000000000499146339837480320000000000000000000000 | ( 8 ; 8 ; 4) | 62355150 | 0.000000003119664623984252200000000000000000000000 | +4.0 | +0.000000000623932924796850480000000000000000000000 | ( 8 ; 9 ; 3) | 27713400 | 0.000000001386517610659667800000000000000000000000 | +6.0 | +0.000000000415955283197900350000000000000000000000 | ( 8 ; 10 ; 2) | 8314020 | 0.000000000415955283197900300000000000000000000000 | +8.0 | +0.000000000166382113279160120000000000000000000000 | ( 8 ; 11 ; 1) | 1511640 | 0.000000000075628233308709151000000000000000000000 | +10.0 | +0.000000000037814116654354576000000000000000000000 | ( 8 ; 12 ; 0) | 125970 | 0.000000000006302352775725761800000000000000000000 | +12.0 | +0.000000000003781411665435457100000000000000000000 | ( 9 ; 0 ; 11) | 167960 | 0.000000000000466840946350056460000000000000000000 | -11.0 | -0.000000000000256762520492531030000000000000000000 | ( 9 ; 1 ; 10) | 1847560 | 0.000000000005135250409850621100000000000000000000 | -9.0 | -0.000000000002310862684432779600000000000000000000 | ( 9 ; 2 ; 9) | 9237800 | 0.000000000025676252049253103000000000000000000000 | -7.0 | -0.000000000008986688217238586700000000000000000000 | ( 9 ; 3 ; 8) | 27713400 | 0.000000000077028756147759322000000000000000000000 | -5.0 | -0.000000000019257189036939830000000000000000000000 | ( 9 ; 4 ; 7) | 55426800 | 0.000000000154057512295518620000000000000000000000 | -3.0 | -0.000000000023108626844327792000000000000000000000 | ( 9 ; 5 ; 6) | 77597520 | 0.000000000215680517213726050000000000000000000000 | -1.0 | -0.000000000010784025860686302000000000000000000000 | ( 9 ; 6 ; 5) | 77597520 | 0.000000000215680517213726030000000000000000000000 | +1.0 | +0.000000000010784025860686301000000000000000000000 | ( 9 ; 7 ; 4) | 55426800 | 0.000000000154057512295518620000000000000000000000 | +3.0 | +0.000000000023108626844327792000000000000000000000 | ( 9 ; 8 ; 3) | 27713400 | 0.000000000077028756147759309000000000000000000000 | +5.0 | +0.000000000019257189036939827000000000000000000000 | ( 9 ; 9 ; 2) | 9237800 | 0.000000000025676252049253103000000000000000000000 | +7.0 | +0.000000000008986688217238586700000000000000000000 | ( 9 ; 10 ; 1) | 1847560 | 0.000000000005135250409850620300000000000000000000 | +9.0 | +0.000000000002310862684432779200000000000000000000 | ( 9 ; 11 ; 0) | 167960 | 0.000000000000466840946350056460000000000000000000 | +11.0 | +0.000000000000256762520492531030000000000000000000 | (10 ; 0 ; 10) | 184756 | 0.000000000000028529168943614561000000000000000000 | -10.0 | -0.000000000000014264584471807281000000000000000000 | (10 ; 1 ; 9) | 1847560 | 0.000000000000285291689436145630000000000000000000 | -8.0 | -0.000000000000114116675774458250000000000000000000 | (10 ; 2 ; 8) | 8314020 | 0.000000000001283812602462655300000000000000000000 | -6.0 | -0.000000000000385143780738796600000000000000000000 | (10 ; 3 ; 7) | 22170720 | 0.000000000003423500273233747900000000000000000000 | -4.0 | -0.000000000000684700054646749620000000000000000000 | (10 ; 4 ; 6) | 38798760 | 0.000000000005991125478159058000000000000000000000 | -2.0 | -0.000000000000599112547815905840000000000000000000 | (10 ; 5 ; 5) | 46558512 | 0.000000000007189350573790870100000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (10 ; 6 ; 4) | 38798760 | 0.000000000005991125478159058900000000000000000000 | +2.0 | +0.000000000000599112547815905840000000000000000000 | (10 ; 7 ; 3) | 22170720 | 0.000000000003423500273233747500000000000000000000 | +4.0 | +0.000000000000684700054646749520000000000000000000 | (10 ; 8 ; 2) | 8314020 | 0.000000000001283812602462655300000000000000000000 | +6.0 | +0.000000000000385143780738796600000000000000000000 | (10 ; 9 ; 1) | 1847560 | 0.000000000000285291689436145630000000000000000000 | +8.0 | +0.000000000000114116675774458250000000000000000000 | (10 ; 10 ; 0) | 184756 | 0.000000000000028529168943614561000000000000000000 | +10.0 | +0.000000000000014264584471807281000000000000000000 | (11 ; 0 ; 9) | 167960 | 0.000000000000001440867118364371900000000000000000 | -9.0 | -0.000000000000000648390203263967350000000000000000 | (11 ; 1 ; 8) | 1511640 | 0.000000000000012967804065279347000000000000000000 | -7.0 | -0.000000000000004538731422847771300000000000000000 | (11 ; 2 ; 7) | 6046560 | 0.000000000000051871216261117383000000000000000000 | -5.0 | -0.000000000000012967804065279346000000000000000000 | (11 ; 3 ; 6) | 14108640 | 0.000000000000121032837942607250000000000000000000 | -3.0 | -0.000000000000018154925691391089000000000000000000 | (11 ; 4 ; 5) | 21162960 | 0.000000000000181549256913910890000000000000000000 | -1.0 | -0.000000000000009077462845695544300000000000000000 | (11 ; 5 ; 4) | 21162960 | 0.000000000000181549256913910890000000000000000000 | +1.0 | +0.000000000000009077462845695544300000000000000000 | (11 ; 6 ; 3) | 14108640 | 0.000000000000121032837942607220000000000000000000 | +3.0 | +0.000000000000018154925691391082000000000000000000 | (11 ; 7 ; 2) | 6046560 | 0.000000000000051871216261117389000000000000000000 | +5.0 | +0.000000000000012967804065279347000000000000000000 | (11 ; 8 ; 1) | 1511640 | 0.000000000000012967804065279347000000000000000000 | +7.0 | +0.000000000000004538731422847771300000000000000000 | (11 ; 9 ; 0) | 167960 | 0.000000000000001440867118364371900000000000000000 | +9.0 | +0.000000000000000648390203263967350000000000000000 | (12 ; 0 ; 8) | 125970 | 0.000000000000000060036129931848819000000000000000 | -8.0 | -0.000000000000000024014451972739528000000000000000 | (12 ; 1 ; 7) | 1007760 | 0.000000000000000480289039454790560000000000000000 | -6.0 | -0.000000000000000144086711836437170000000000000000 | (12 ; 2 ; 6) | 3527160 | 0.000000000000001681011638091766900000000000000000 | -4.0 | -0.000000000000000336202327618353390000000000000000 | (12 ; 3 ; 5) | 7054320 | 0.000000000000003362023276183534300000000000000000 | -2.0 | -0.000000000000000336202327618353440000000000000000 | (12 ; 4 ; 4) | 8817900 | 0.000000000000004202529095229418300000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (12 ; 5 ; 3) | 7054320 | 0.000000000000003362023276183533900000000000000000 | +2.0 | +0.000000000000000336202327618353390000000000000000 | (12 ; 6 ; 2) | 3527160 | 0.000000000000001681011638091767100000000000000000 | +4.0 | +0.000000000000000336202327618353440000000000000000 | (12 ; 7 ; 1) | 1007760 | 0.000000000000000480289039454790560000000000000000 | +6.0 | +0.000000000000000144086711836437170000000000000000 | (12 ; 8 ; 0) | 125970 | 0.000000000000000060036129931848819000000000000000 | +8.0 | +0.000000000000000024014451972739528000000000000000 | (13 ; 0 ; 7) | 77520 | 0.000000000000000002052517262627310100000000000000 | -7.0 | -0.000000000000000000718381041919558520000000000000 | (13 ; 1 ; 6) | 542640 | 0.000000000000000014367620838391170000000000000000 | -5.0 | -0.000000000000000003591905209597792600000000000000 | (13 ; 2 ; 5) | 1627920 | 0.000000000000000043102862515173520000000000000000 | -3.0 | -0.000000000000000006465429377276028200000000000000 | (13 ; 3 ; 4) | 2713200 | 0.000000000000000071838104191955867000000000000000 | -1.0 | -0.000000000000000003591905209597793300000000000000 | (13 ; 4 ; 3) | 2713200 | 0.000000000000000071838104191955855000000000000000 | +1.0 | +0.000000000000000003591905209597792600000000000000 | (13 ; 5 ; 2) | 1627920 | 0.000000000000000043102862515173514000000000000000 | +3.0 | +0.000000000000000006465429377276027400000000000000 | (13 ; 6 ; 1) | 542640 | 0.000000000000000014367620838391173000000000000000 | +5.0 | +0.000000000000000003591905209597793300000000000000 | (13 ; 7 ; 0) | 77520 | 0.000000000000000002052517262627310100000000000000 | +7.0 | +0.000000000000000000718381041919558520000000000000 | (14 ; 0 ; 6) | 38760 | 0.000000000000000000057014368406314167000000000000 | -6.0 | -0.000000000000000000017104310521894251000000000000 | (14 ; 1 ; 5) | 232560 | 0.000000000000000000342086210437885050000000000000 | -4.0 | -0.000000000000000000068417242087577005000000000000 | (14 ; 2 ; 4) | 581400 | 0.000000000000000000855215526094712480000000000000 | -2.0 | -0.000000000000000000085521552609471250000000000000 | (14 ; 3 ; 3) | 775200 | 0.000000000000000001140287368126283400000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (14 ; 4 ; 2) | 581400 | 0.000000000000000000855215526094712380000000000000 | +2.0 | +0.000000000000000000085521552609471238000000000000 | (14 ; 5 ; 1) | 232560 | 0.000000000000000000342086210437885000000000000000 | +4.0 | +0.000000000000000000068417242087577005000000000000 | (14 ; 6 ; 0) | 38760 | 0.000000000000000000057014368406314167000000000000 | +6.0 | +0.000000000000000000017104310521894251000000000000 | (15 ; 0 ; 5) | 15504 | 0.000000000000000000001266985964584759500000000000 | -5.0 | -0.000000000000000000000316746491146189870000000000 | (15 ; 1 ; 4) | 77520 | 0.000000000000000000006334929822923797200000000000 | -3.0 | -0.000000000000000000000950239473438569660000000000 | (15 ; 2 ; 3) | 155040 | 0.000000000000000000012669859645847596000000000000 | -1.0 | -0.000000000000000000000633492982292379840000000000 | (15 ; 3 ; 2) | 155040 | 0.000000000000000000012669859645847594000000000000 | +1.0 | +0.000000000000000000000633492982292379740000000000 | (15 ; 4 ; 1) | 77520 | 0.000000000000000000006334929822923798000000000000 | +3.0 | +0.000000000000000000000950239473438569660000000000 | (15 ; 5 ; 0) | 15504 | 0.000000000000000000001266985964584759500000000000 | +5.0 | +0.000000000000000000000316746491146189870000000000 | (16 ; 0 ; 4) | 4845 | 0.000000000000000000000021996284107374294000000000 | -4.0 | -0.000000000000000000000004399256821474859200000000 | (16 ; 1 ; 3) | 19380 | 0.000000000000000000000087985136429497177000000000 | -2.0 | -0.000000000000000000000008798513642949718300000000 | (16 ; 2 ; 2) | 29070 | 0.000000000000000000000131977704644245780000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (16 ; 3 ; 1) | 19380 | 0.000000000000000000000087985136429497177000000000 | +2.0 | +0.000000000000000000000008798513642949718300000000 | (16 ; 4 ; 0) | 4845 | 0.000000000000000000000021996284107374294000000000 | +4.0 | +0.000000000000000000000004399256821474859200000000 | (17 ; 0 ; 3) | 1140 | 0.000000000000000000000000287533125586592110000000 | -3.0 | -0.000000000000000000000000043129968837988816000000 | (17 ; 1 ; 2) | 3420 | 0.000000000000000000000000862599376759776370000000 | -1.0 | -0.000000000000000000000000043129968837988816000000 | (17 ; 2 ; 1) | 3420 | 0.000000000000000000000000862599376759776370000000 | +1.0 | +0.000000000000000000000000043129968837988816000000 | (17 ; 3 ; 0) | 1140 | 0.000000000000000000000000287533125586592110000000 | +3.0 | +0.000000000000000000000000043129968837988816000000 | (18 ; 0 ; 2) | 190 | 0.000000000000000000000000002662343755431408000000 | -2.0 | -0.000000000000000000000000000266234375543140800000 | (18 ; 1 ; 1) | 380 | 0.000000000000000000000000005324687510862816000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (18 ; 2 ; 0) | 190 | 0.000000000000000000000000002662343755431408000000 | +2.0 | +0.000000000000000000000000000266234375543140800000 | (19 ; 0 ; 1) | 20 | 0.000000000000000000000000000015569261727669054000 | -1.0 | -0.000000000000000000000000000000778463086383452730 | (19 ; 1 ; 0) | 20 | 0.000000000000000000000000000015569261727669054000 | +1.0 | +0.000000000000000000000000000000778463086383452730 | (20 ; 0 ; 0) | 1 | 0.000000000000000000000000000000043247949243525151 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ============== | ============ | ================================================== | ===== | =================================================== | Probabilité du gain : +0.441044184688224540 Probabilité du nul : +0.117911630623551810 Probabilité de la perte : +0.441044184688224590 Cumul des Probabilités : +1.000000000000001100 Espérance mathématique : +0.000000000000000004
19/12/2012 3:10 am
(désolé, mais je n'ai pas pu tout mettre dans le même message)
Et voici la roulette européenne, toujours avec la loi multinomiale :
Triplets | Combinaisons | Probabilités | Bilan | Espérances Mathématiques | ============== | ============ | ================================================== | ===== | =================================================== | ( 0 ; 0 ; 20) | 1 | 0.000000551335072831166730000000000000000000000000 | -20.0 | -0.000000551335072831166730000000000000000000000000 | ( 0 ; 1 ; 19) | 20 | 0.000011026701456623334000000000000000000000000000 | -18.0 | -0.000009924031310961000900000000000000000000000000 | ( 0 ; 2 ; 18) | 190 | 0.000104753663837921680000000000000000000000000000 | -16.0 | -0.000083802931070337347000000000000000000000000000 | ( 0 ; 3 ; 17) | 1140 | 0.000628521983027530090000000000000000000000000000 | -14.0 | -0.000439965388119271050000000000000000000000000000 | ( 0 ; 4 ; 16) | 4845 | 0.002671218427867002600000000000000000000000000000 | -12.0 | -0.001602731056720201500000000000000000000000000000 | ( 0 ; 5 ; 15) | 15504 | 0.008547898969174409100000000000000000000000000000 | -10.0 | -0.004273949484587204500000000000000000000000000000 | ( 0 ; 6 ; 14) | 38760 | 0.021369747422936021000000000000000000000000000000 | -8.0 | -0.008547898969174409100000000000000000000000000000 | ( 0 ; 7 ; 13) | 77520 | 0.042739494845872042000000000000000000000000000000 | -6.0 | -0.012821848453761612000000000000000000000000000000 | ( 0 ; 8 ; 12) | 125970 | 0.069451679124542073000000000000000000000000000000 | -4.0 | -0.013890335824908414000000000000000000000000000000 | ( 0 ; 9 ; 11) | 167960 | 0.092602238832722755000000000000000000000000000000 | -2.0 | -0.009260223883272276200000000000000000000000000000 | ( 0 ; 10 ; 10) | 184756 | 0.101862462715995030000000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 0 ; 11 ; 9) | 167960 | 0.092602238832722755000000000000000000000000000000 | +2.0 | +0.009260223883272276200000000000000000000000000000 | ( 0 ; 12 ; 8) | 125970 | 0.069451679124542073000000000000000000000000000000 | +4.0 | +0.013890335824908414000000000000000000000000000000 | ( 0 ; 13 ; 7) | 77520 | 0.042739494845872042000000000000000000000000000000 | +6.0 | +0.012821848453761612000000000000000000000000000000 | ( 0 ; 14 ; 6) | 38760 | 0.021369747422936021000000000000000000000000000000 | +8.0 | +0.008547898969174409100000000000000000000000000000 | ( 0 ; 15 ; 5) | 15504 | 0.008547898969174409100000000000000000000000000000 | +10.0 | +0.004273949484587204500000000000000000000000000000 | ( 0 ; 16 ; 4) | 4845 | 0.002671218427867002600000000000000000000000000000 | +12.0 | +0.001602731056720201500000000000000000000000000000 | ( 0 ; 17 ; 3) | 1140 | 0.000628521983027530090000000000000000000000000000 | +14.0 | +0.000439965388119271050000000000000000000000000000 | ( 0 ; 18 ; 2) | 190 | 0.000104753663837921680000000000000000000000000000 | +16.0 | +0.000083802931070337347000000000000000000000000000 | ( 0 ; 19 ; 1) | 20 | 0.000011026701456623334000000000000000000000000000 | +18.0 | +0.000009924031310961000900000000000000000000000000 | ( 0 ; 20 ; 0) | 1 | 0.000000551335072831166730000000000000000000000000 | +20.0 | +0.000000551335072831166730000000000000000000000000 | ( 1 ; 0 ; 19) | 20 | 0.000000612594525367962980000000000000000000000000 | -20.0 | -0.000000612594525367962980000000000000000000000000 | ( 1 ; 1 ; 18) | 380 | 0.000011639295981991295000000000000000000000000000 | -18.0 | -0.000010475366383792165000000000000000000000000000 | ( 1 ; 2 ; 17) | 3420 | 0.000104753663837921660000000000000000000000000000 | -16.0 | -0.000083802931070337333000000000000000000000000000 | ( 1 ; 3 ; 16) | 19380 | 0.000593604095081556200000000000000000000000000000 | -14.0 | -0.000415522866557089360000000000000000000000000000 | ( 1 ; 4 ; 15) | 77520 | 0.002374416380326224800000000000000000000000000000 | -12.0 | -0.001424649828195734900000000000000000000000000000 | ( 1 ; 5 ; 14) | 232560 | 0.007123249140978673900000000000000000000000000000 | -10.0 | -0.003561624570489337000000000000000000000000000000 | ( 1 ; 6 ; 13) | 542640 | 0.016620914662283572000000000000000000000000000000 | -8.0 | -0.006648365864913428900000000000000000000000000000 | ( 1 ; 7 ; 12) | 1007760 | 0.030867412944240915000000000000000000000000000000 | -6.0 | -0.009260223883272274500000000000000000000000000000 | ( 1 ; 8 ; 11) | 1511640 | 0.046301119416361385000000000000000000000000000000 | -4.0 | -0.009260223883272276200000000000000000000000000000 | ( 1 ; 9 ; 10) | 1847560 | 0.056590257064441686000000000000000000000000000000 | -2.0 | -0.005659025706444168600000000000000000000000000000 | ( 1 ; 10 ; 9) | 1847560 | 0.056590257064441686000000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 1 ; 11 ; 8) | 1511640 | 0.046301119416361385000000000000000000000000000000 | +2.0 | +0.004630111941636138100000000000000000000000000000 | ( 1 ; 12 ; 7) | 1007760 | 0.030867412944240918000000000000000000000000000000 | +4.0 | +0.006173482588848183900000000000000000000000000000 | ( 1 ; 13 ; 6) | 542640 | 0.016620914662283572000000000000000000000000000000 | +6.0 | +0.004986274398685071700000000000000000000000000000 | ( 1 ; 14 ; 5) | 232560 | 0.007123249140978673900000000000000000000000000000 | +8.0 | +0.002849299656391469400000000000000000000000000000 | ( 1 ; 15 ; 4) | 77520 | 0.002374416380326224800000000000000000000000000000 | +10.0 | +0.001187208190163112400000000000000000000000000000 | ( 1 ; 16 ; 3) | 19380 | 0.000593604095081556200000000000000000000000000000 | +12.0 | +0.000356162457048933730000000000000000000000000000 | ( 1 ; 17 ; 2) | 3420 | 0.000104753663837921660000000000000000000000000000 | +14.0 | +0.000073327564686545161000000000000000000000000000 | ( 1 ; 18 ; 1) | 380 | 0.000011639295981991296000000000000000000000000000 | +16.0 | +0.000009311436785593036800000000000000000000000000 | ( 1 ; 19 ; 0) | 20 | 0.000000612594525367962980000000000000000000000000 | +18.0 | +0.000000551335072831166730000000000000000000000000 | ( 2 ; 0 ; 18) | 190 | 0.000000323313777277536050000000000000000000000000 | -20.0 | -0.000000323313777277536050000000000000000000000000 | ( 2 ; 1 ; 17) | 3420 | 0.000005819647990995648200000000000000000000000000 | -18.0 | -0.000005237683191896083300000000000000000000000000 | ( 2 ; 2 ; 16) | 29070 | 0.000049467007923463012000000000000000000000000000 | -16.0 | -0.000039573606338770407000000000000000000000000000 | ( 2 ; 3 ; 15) | 155040 | 0.000263824042258469420000000000000000000000000000 | -14.0 | -0.000184676829580928600000000000000000000000000000 | ( 2 ; 4 ; 14) | 581400 | 0.000989340158469260290000000000000000000000000000 | -12.0 | -0.000593604095081556200000000000000000000000000000 | ( 2 ; 5 ; 13) | 1627920 | 0.002770152443713928600000000000000000000000000000 | -10.0 | -0.001385076221856964300000000000000000000000000000 | ( 2 ; 6 ; 12) | 3527160 | 0.006001996961380179400000000000000000000000000000 | -8.0 | -0.002400798784552071700000000000000000000000000000 | ( 2 ; 7 ; 11) | 6046560 | 0.010289137648080307000000000000000000000000000000 | -6.0 | -0.003086741294424091900000000000000000000000000000 | ( 2 ; 8 ; 10) | 8314020 | 0.014147564266110422000000000000000000000000000000 | -4.0 | -0.002829512853222084300000000000000000000000000000 | ( 2 ; 9 ; 9) | 9237800 | 0.015719515851233802000000000000000000000000000000 | -2.0 | -0.001571951585123380100000000000000000000000000000 | ( 2 ; 10 ; 8) | 8314020 | 0.014147564266110422000000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 2 ; 11 ; 7) | 6046560 | 0.010289137648080307000000000000000000000000000000 | +2.0 | +0.001028913764808030700000000000000000000000000000 | ( 2 ; 12 ; 6) | 3527160 | 0.006001996961380179400000000000000000000000000000 | +4.0 | +0.001200399392276035900000000000000000000000000000 | ( 2 ; 13 ; 5) | 1627920 | 0.002770152443713928600000000000000000000000000000 | +6.0 | +0.000831045733114178610000000000000000000000000000 | ( 2 ; 14 ; 4) | 581400 | 0.000989340158469260290000000000000000000000000000 | +8.0 | +0.000395736063387704100000000000000000000000000000 | ( 2 ; 15 ; 3) | 155040 | 0.000263824042258469420000000000000000000000000000 | +10.0 | +0.000131912021129234710000000000000000000000000000 | ( 2 ; 16 ; 2) | 29070 | 0.000049467007923463019000000000000000000000000000 | +12.0 | +0.000029680204754077812000000000000000000000000000 | ( 2 ; 17 ; 1) | 3420 | 0.000005819647990995648200000000000000000000000000 | +14.0 | +0.000004073753593696953500000000000000000000000000 | ( 2 ; 18 ; 0) | 190 | 0.000000323313777277536050000000000000000000000000 | +16.0 | +0.000000258651021822028850000000000000000000000000 | ( 3 ; 0 ; 17) | 1140 | 0.000000107771259092512000000000000000000000000000 | -20.0 | -0.000000107771259092512000000000000000000000000000 | ( 3 ; 1 ; 16) | 19380 | 0.000001832111404572704000000000000000000000000000 | -18.0 | -0.000001648900264115433600000000000000000000000000 | ( 3 ; 2 ; 15) | 155040 | 0.000014656891236581632000000000000000000000000000 | -16.0 | -0.000011725512989265306000000000000000000000000000 | ( 3 ; 3 ; 14) | 775200 | 0.000073284456182908165000000000000000000000000000 | -14.0 | -0.000051299119328035718000000000000000000000000000 | ( 3 ; 4 ; 13) | 2713200 | 0.000256495596640178590000000000000000000000000000 | -12.0 | -0.000153897357984107150000000000000000000000000000 | ( 3 ; 5 ; 12) | 7054320 | 0.000666888551264464230000000000000000000000000000 | -10.0 | -0.000333444275632232110000000000000000000000000000 | ( 3 ; 6 ; 11) | 14108640 | 0.001333777102528928500000000000000000000000000000 | -8.0 | -0.000533510841011571430000000000000000000000000000 | ( 3 ; 7 ; 10) | 22170720 | 0.002095935446831173300000000000000000000000000000 | -6.0 | -0.000628780634049351980000000000000000000000000000 | ( 3 ; 8 ; 9) | 27713400 | 0.002619919308538966600000000000000000000000000000 | -4.0 | -0.000523983861707793340000000000000000000000000000 | ( 3 ; 9 ; 8) | 27713400 | 0.002619919308538966600000000000000000000000000000 | -2.0 | -0.000261991930853896670000000000000000000000000000 | ( 3 ; 10 ; 7) | 22170720 | 0.002095935446831173300000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 3 ; 11 ; 6) | 14108640 | 0.001333777102528928500000000000000000000000000000 | +2.0 | +0.000133377710252892860000000000000000000000000000 | ( 3 ; 12 ; 5) | 7054320 | 0.000666888551264464230000000000000000000000000000 | +4.0 | +0.000133377710252892860000000000000000000000000000 | ( 3 ; 13 ; 4) | 2713200 | 0.000256495596640178590000000000000000000000000000 | +6.0 | +0.000076948678992053577000000000000000000000000000 | ( 3 ; 14 ; 3) | 775200 | 0.000073284456182908165000000000000000000000000000 | +8.0 | +0.000029313782473163265000000000000000000000000000 | ( 3 ; 15 ; 2) | 155040 | 0.000014656891236581631000000000000000000000000000 | +10.0 | +0.000007328445618290815300000000000000000000000000 | ( 3 ; 16 ; 1) | 19380 | 0.000001832111404572704000000000000000000000000000 | +12.0 | +0.000001099266842743622500000000000000000000000000 | ( 3 ; 17 ; 0) | 1140 | 0.000000107771259092512000000000000000000000000000 | +14.0 | +0.000000075439881364758405000000000000000000000000 | ( 4 ; 0 ; 16) | 4845 | 0.000000025445991730176449000000000000000000000000 | -20.0 | -0.000000025445991730176449000000000000000000000000 | ( 4 ; 1 ; 15) | 77520 | 0.000000407135867682823190000000000000000000000000 | -18.0 | -0.000000366422280914540870000000000000000000000000 | ( 4 ; 2 ; 14) | 581400 | 0.000003053519007621173400000000000000000000000000 | -16.0 | -0.000002442815206096938700000000000000000000000000 | ( 4 ; 3 ; 13) | 2713200 | 0.000014249755368898811000000000000000000000000000 | -14.0 | -0.000009974828758229167400000000000000000000000000 | ( 4 ; 4 ; 12) | 8817900 | 0.000046311704948921134000000000000000000000000000 | -12.0 | -0.000027787022969352681000000000000000000000000000 | ( 4 ; 5 ; 11) | 21162960 | 0.000111148091877410710000000000000000000000000000 | -10.0 | -0.000055574045938705355000000000000000000000000000 | ( 4 ; 6 ; 10) | 38798760 | 0.000203771501775252960000000000000000000000000000 | -8.0 | -0.000081508600710101189000000000000000000000000000 | ( 4 ; 7 ; 9) | 55426800 | 0.000291102145393218570000000000000000000000000000 | -6.0 | -0.000087330643617965570000000000000000000000000000 | ( 4 ; 8 ; 8) | 62355150 | 0.000327489913567370880000000000000000000000000000 | -4.0 | -0.000065497982713474181000000000000000000000000000 | ( 4 ; 9 ; 7) | 55426800 | 0.000291102145393218520000000000000000000000000000 | -2.0 | -0.000029110214539321852000000000000000000000000000 | ( 4 ; 10 ; 6) | 38798760 | 0.000203771501775252960000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 4 ; 11 ; 5) | 21162960 | 0.000111148091877410720000000000000000000000000000 | +2.0 | +0.000011114809187741072000000000000000000000000000 | ( 4 ; 12 ; 4) | 8817900 | 0.000046311704948921134000000000000000000000000000 | +4.0 | +0.000009262340989784226900000000000000000000000000 | ( 4 ; 13 ; 3) | 2713200 | 0.000014249755368898811000000000000000000000000000 | +6.0 | +0.000004274926610669643200000000000000000000000000 | ( 4 ; 14 ; 2) | 581400 | 0.000003053519007621173400000000000000000000000000 | +8.0 | +0.000001221407603048469400000000000000000000000000 | ( 4 ; 15 ; 1) | 77520 | 0.000000407135867682823140000000000000000000000000 | +10.0 | +0.000000203567933841411570000000000000000000000000 | ( 4 ; 16 ; 0) | 4845 | 0.000000025445991730176449000000000000000000000000 | +12.0 | +0.000000015267595038105870000000000000000000000000 | ( 5 ; 0 ; 15) | 15504 | 0.000000004523731863142479100000000000000000000000 | -20.0 | -0.000000004523731863142479100000000000000000000000 | ( 5 ; 1 ; 14) | 232560 | 0.000000067855977947137198000000000000000000000000 | -18.0 | -0.000000061070380152423479000000000000000000000000 | ( 5 ; 2 ; 13) | 1627920 | 0.000000474991845629960390000000000000000000000000 | -16.0 | -0.000000379993476503968310000000000000000000000000 | ( 5 ; 3 ; 12) | 7054320 | 0.000002058297997729827800000000000000000000000000 | -14.0 | -0.000001440808598410879500000000000000000000000000 | ( 5 ; 4 ; 11) | 21162960 | 0.000006174893993189484300000000000000000000000000 | -12.0 | -0.000003704936395913690500000000000000000000000000 | ( 5 ; 5 ; 10) | 46558512 | 0.000013584766785016867000000000000000000000000000 | -10.0 | -0.000006792383392508433500000000000000000000000000 | ( 5 ; 6 ; 9) | 77597520 | 0.000022641277975028109000000000000000000000000000 | -8.0 | -0.000009056511190011244100000000000000000000000000 | ( 5 ; 7 ; 8) | 99768240 | 0.000029110214539321855000000000000000000000000000 | -6.0 | -0.000008733064361796556600000000000000000000000000 | ( 5 ; 8 ; 7) | 99768240 | 0.000029110214539321852000000000000000000000000000 | -4.0 | -0.000005822042907864370200000000000000000000000000 | ( 5 ; 9 ; 6) | 77597520 | 0.000022641277975028109000000000000000000000000000 | -2.0 | -0.000002264127797502811000000000000000000000000000 | ( 5 ; 10 ; 5) | 46558512 | 0.000013584766785016867000000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 5 ; 11 ; 4) | 21162960 | 0.000006174893993189484300000000000000000000000000 | +2.0 | +0.000000617489399318948450000000000000000000000000 | ( 5 ; 12 ; 3) | 7054320 | 0.000002058297997729827800000000000000000000000000 | +4.0 | +0.000000411659599545965560000000000000000000000000 | ( 5 ; 13 ; 2) | 1627920 | 0.000000474991845629960340000000000000000000000000 | +6.0 | +0.000000142497553688988090000000000000000000000000 | ( 5 ; 14 ; 1) | 232560 | 0.000000067855977947137198000000000000000000000000 | +8.0 | +0.000000027142391178854879000000000000000000000000 | ( 5 ; 15 ; 0) | 15504 | 0.000000004523731863142479100000000000000000000000 | +10.0 | +0.000000002261865931571239500000000000000000000000 | ( 6 ; 0 ; 14) | 38760 | 0.000000000628296092103122160000000000000000000000 | -20.0 | -0.000000000628296092103122160000000000000000000000 | ( 6 ; 1 ; 13) | 542640 | 0.000000008796145289443711100000000000000000000000 | -18.0 | -0.000000007916530760499339800000000000000000000000 | ( 6 ; 2 ; 12) | 3527160 | 0.000000057174944381384111000000000000000000000000 | -16.0 | -0.000000045739955505107286000000000000000000000000 | ( 6 ; 3 ; 11) | 14108640 | 0.000000228699777525536420000000000000000000000000 | -14.0 | -0.000000160089844267875480000000000000000000000000 | ( 6 ; 4 ; 10) | 38798760 | 0.000000628924388195225300000000000000000000000000 | -12.0 | -0.000000377354632917135190000000000000000000000000 | ( 6 ; 5 ; 9) | 77597520 | 0.000001257848776390450600000000000000000000000000 | -10.0 | -0.000000628924388195225300000000000000000000000000 | ( 6 ; 6 ; 8) | 116396280 | 0.000001886773164585675600000000000000000000000000 | -8.0 | -0.000000754709265834270270000000000000000000000000 | ( 6 ; 7 ; 7) | 133024320 | 0.000002156312188097915100000000000000000000000000 | -6.0 | -0.000000646893656429374520000000000000000000000000 | ( 6 ; 8 ; 6) | 116396280 | 0.000001886773164585675600000000000000000000000000 | -4.0 | -0.000000377354632917135140000000000000000000000000 | ( 6 ; 9 ; 5) | 77597520 | 0.000001257848776390450600000000000000000000000000 | -2.0 | -0.000000125784877639045050000000000000000000000000 | ( 6 ; 10 ; 4) | 38798760 | 0.000000628924388195225300000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 6 ; 11 ; 3) | 14108640 | 0.000000228699777525536450000000000000000000000000 | +2.0 | +0.000000022869977752553643000000000000000000000000 | ( 6 ; 12 ; 2) | 3527160 | 0.000000057174944381384105000000000000000000000000 | +4.0 | +0.000000011434988876276822000000000000000000000000 | ( 6 ; 13 ; 1) | 542640 | 0.000000008796145289443709400000000000000000000000 | +6.0 | +0.000000002638843586833113000000000000000000000000 | ( 6 ; 14 ; 0) | 38760 | 0.000000000628296092103122160000000000000000000000 | +8.0 | +0.000000000251318436841248850000000000000000000000 | ( 7 ; 0 ; 13) | 77520 | 0.000000000069810676900346914000000000000000000000 | -20.0 | -0.000000000069810676900346914000000000000000000000 | ( 7 ; 1 ; 12) | 1007760 | 0.000000000907538799704509870000000000000000000000 | -18.0 | -0.000000000816784919734058890000000000000000000000 | ( 7 ; 2 ; 11) | 6046560 | 0.000000005445232798227058200000000000000000000000 | -16.0 | -0.000000004356186238581646900000000000000000000000 | ( 7 ; 3 ; 10) | 22170720 | 0.000000019965853593499215000000000000000000000000 | -14.0 | -0.000000013976097515449451000000000000000000000000 | ( 7 ; 4 ; 9) | 55426800 | 0.000000049914633983748041000000000000000000000000 | -12.0 | -0.000000029948780390248826000000000000000000000000 | ( 7 ; 5 ; 8) | 99768240 | 0.000000089846341170746465000000000000000000000000 | -10.0 | -0.000000044923170585373233000000000000000000000000 | ( 7 ; 6 ; 7) | 133024320 | 0.000000119795121560995300000000000000000000000000 | -8.0 | -0.000000047918048624398121000000000000000000000000 | ( 7 ; 7 ; 6) | 133024320 | 0.000000119795121560995280000000000000000000000000 | -6.0 | -0.000000035938536468298582000000000000000000000000 | ( 7 ; 8 ; 5) | 99768240 | 0.000000089846341170746465000000000000000000000000 | -4.0 | -0.000000017969268234149294000000000000000000000000 | ( 7 ; 9 ; 4) | 55426800 | 0.000000049914633983748041000000000000000000000000 | -2.0 | -0.000000004991463398374803800000000000000000000000 | ( 7 ; 10 ; 3) | 22170720 | 0.000000019965853593499215000000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 7 ; 11 ; 2) | 6046560 | 0.000000005445232798227058200000000000000000000000 | +2.0 | +0.000000000544523279822705860000000000000000000000 | ( 7 ; 12 ; 1) | 1007760 | 0.000000000907538799704509870000000000000000000000 | +4.0 | +0.000000000181507759940901980000000000000000000000 | ( 7 ; 13 ; 0) | 77520 | 0.000000000069810676900346914000000000000000000000 | +6.0 | +0.000000000020943203070104075000000000000000000000 | ( 8 ; 0 ; 12) | 125970 | 0.000000000006302352775725761800000000000000000000 | -20.0 | -0.000000000006302352775725761800000000000000000000 | ( 8 ; 1 ; 11) | 1511640 | 0.000000000075628233308709139000000000000000000000 | -18.0 | -0.000000000068065409977838220000000000000000000000 | ( 8 ; 2 ; 10) | 8314020 | 0.000000000415955283197900350000000000000000000000 | -16.0 | -0.000000000332764226558320280000000000000000000000 | ( 8 ; 3 ; 9) | 27713400 | 0.000000001386517610659667600000000000000000000000 | -14.0 | -0.000000000970562327461767390000000000000000000000 | ( 8 ; 4 ; 8) | 62355150 | 0.000000003119664623984252200000000000000000000000 | -12.0 | -0.000000001871798774390551200000000000000000000000 | ( 8 ; 5 ; 7) | 99768240 | 0.000000004991463398374803000000000000000000000000 | -10.0 | -0.000000002495731699187401500000000000000000000000 | ( 8 ; 6 ; 6) | 116396280 | 0.000000005823373964770603900000000000000000000000 | -8.0 | -0.000000002329349585908241600000000000000000000000 | ( 8 ; 7 ; 5) | 99768240 | 0.000000004991463398374803000000000000000000000000 | -6.0 | -0.000000001497439019512440900000000000000000000000 | ( 8 ; 8 ; 4) | 62355150 | 0.000000003119664623984252200000000000000000000000 | -4.0 | -0.000000000623932924796850480000000000000000000000 | ( 8 ; 9 ; 3) | 27713400 | 0.000000001386517610659667800000000000000000000000 | -2.0 | -0.000000000138651761065966780000000000000000000000 | ( 8 ; 10 ; 2) | 8314020 | 0.000000000415955283197900300000000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 8 ; 11 ; 1) | 1511640 | 0.000000000075628233308709151000000000000000000000 | +2.0 | +0.000000000007562823330870915800000000000000000000 | ( 8 ; 12 ; 0) | 125970 | 0.000000000006302352775725761800000000000000000000 | +4.0 | +0.000000000001260470555145152400000000000000000000 | ( 9 ; 0 ; 11) | 167960 | 0.000000000000466840946350056460000000000000000000 | -20.0 | -0.000000000000466840946350056460000000000000000000 | ( 9 ; 1 ; 10) | 1847560 | 0.000000000005135250409850621100000000000000000000 | -18.0 | -0.000000000004621725368865559200000000000000000000 | ( 9 ; 2 ; 9) | 9237800 | 0.000000000025676252049253103000000000000000000000 | -16.0 | -0.000000000020541001639402481000000000000000000000 | ( 9 ; 3 ; 8) | 27713400 | 0.000000000077028756147759322000000000000000000000 | -14.0 | -0.000000000053920129303431526000000000000000000000 | ( 9 ; 4 ; 7) | 55426800 | 0.000000000154057512295518620000000000000000000000 | -12.0 | -0.000000000092434507377311168000000000000000000000 | ( 9 ; 5 ; 6) | 77597520 | 0.000000000215680517213726050000000000000000000000 | -10.0 | -0.000000000107840258606863030000000000000000000000 | ( 9 ; 6 ; 5) | 77597520 | 0.000000000215680517213726030000000000000000000000 | -8.0 | -0.000000000086272206885490406000000000000000000000 | ( 9 ; 7 ; 4) | 55426800 | 0.000000000154057512295518620000000000000000000000 | -6.0 | -0.000000000046217253688655584000000000000000000000 | ( 9 ; 8 ; 3) | 27713400 | 0.000000000077028756147759309000000000000000000000 | -4.0 | -0.000000000015405751229551862000000000000000000000 | ( 9 ; 9 ; 2) | 9237800 | 0.000000000025676252049253103000000000000000000000 | -2.0 | -0.000000000002567625204925310100000000000000000000 | ( 9 ; 10 ; 1) | 1847560 | 0.000000000005135250409850620300000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | ( 9 ; 11 ; 0) | 167960 | 0.000000000000466840946350056460000000000000000000 | +2.0 | +0.000000000000046684094635005644000000000000000000 | (10 ; 0 ; 10) | 184756 | 0.000000000000028529168943614561000000000000000000 | -20.0 | -0.000000000000028529168943614561000000000000000000 | (10 ; 1 ; 9) | 1847560 | 0.000000000000285291689436145630000000000000000000 | -18.0 | -0.000000000000256762520492531080000000000000000000 | (10 ; 2 ; 8) | 8314020 | 0.000000000001283812602462655300000000000000000000 | -16.0 | -0.000000000001027050081970124100000000000000000000 | (10 ; 3 ; 7) | 22170720 | 0.000000000003423500273233747900000000000000000000 | -14.0 | -0.000000000002396450191263623400000000000000000000 | (10 ; 4 ; 6) | 38798760 | 0.000000000005991125478159058000000000000000000000 | -12.0 | -0.000000000003594675286895434700000000000000000000 | (10 ; 5 ; 5) | 46558512 | 0.000000000007189350573790870100000000000000000000 | -10.0 | -0.000000000003594675286895435100000000000000000000 | (10 ; 6 ; 4) | 38798760 | 0.000000000005991125478159058900000000000000000000 | -8.0 | -0.000000000002396450191263623400000000000000000000 | (10 ; 7 ; 3) | 22170720 | 0.000000000003423500273233747500000000000000000000 | -6.0 | -0.000000000001027050081970124300000000000000000000 | (10 ; 8 ; 2) | 8314020 | 0.000000000001283812602462655300000000000000000000 | -4.0 | -0.000000000000256762520492531030000000000000000000 | (10 ; 9 ; 1) | 1847560 | 0.000000000000285291689436145630000000000000000000 | -2.0 | -0.000000000000028529168943614561000000000000000000 | (10 ; 10 ; 0) | 184756 | 0.000000000000028529168943614561000000000000000000 | +0.0 | +0.000000000000000000000000000000000000000000000000 | (11 ; 0 ; 9) | 167960 | 0.000000000000001440867118364371900000000000000000 | -20.0 | -0.000000000000001440867118364371900000000000000000 | (11 ; 1 ; 8) | 1511640 | 0.000000000000012967804065279347000000000000000000 | -18.0 | -0.000000000000011671023658751412000000000000000000 | (11 ; 2 ; 7) | 6046560 | 0.000000000000051871216261117383000000000000000000 | -16.0 | -0.000000000000041496973008893904000000000000000000 | (11 ; 3 ; 6) | 14108640 | 0.000000000000121032837942607250000000000000000000 | -14.0 | -0.000000000000084722986559825071000000000000000000 | (11 ; 4 ; 5) | 21162960 | 0.000000000000181549256913910890000000000000000000 | -12.0 | -0.000000000000108929554148346530000000000000000000 | (11 ; 5 ; 4) | 21162960 | 0.000000000000181549256913910890000000000000000000 | -10.0 | -0.000000000000090774628456955443000000000000000000 | (11 ; 6 ; 3) | 14108640 | 0.000000000000121032837942607220000000000000000000 | -8.0 | -0.000000000000048413135177042888000000000000000000 | (11 ; 7 ; 2) | 6046560 | 0.000000000000051871216261117389000000000000000000 | -6.0 | -0.000000000000015561364878335216000000000000000000 | (11 ; 8 ; 1) | 1511640 | 0.000000000000012967804065279347000000000000000000 | -4.0 | -0.000000000000002593560813055869400000000000000000 | (11 ; 9 ; 0) | 167960 | 0.000000000000001440867118364371900000000000000000 | -2.0 | -0.000000000000000144086711836437190000000000000000 | (12 ; 0 ; 8) | 125970 | 0.000000000000000060036129931848819000000000000000 | -20.0 | -0.000000000000000060036129931848819000000000000000 | (12 ; 1 ; 7) | 1007760 | 0.000000000000000480289039454790560000000000000000 | -18.0 | -0.000000000000000432260135509311500000000000000000 | (12 ; 2 ; 6) | 3527160 | 0.000000000000001681011638091766900000000000000000 | -16.0 | -0.000000000000001344809310473413600000000000000000 | (12 ; 3 ; 5) | 7054320 | 0.000000000000003362023276183534300000000000000000 | -14.0 | -0.000000000000002353416293328474100000000000000000 | (12 ; 4 ; 4) | 8817900 | 0.000000000000004202529095229418300000000000000000 | -12.0 | -0.000000000000002521517457137651000000000000000000 | (12 ; 5 ; 3) | 7054320 | 0.000000000000003362023276183533900000000000000000 | -10.0 | -0.000000000000001681011638091766900000000000000000 | (12 ; 6 ; 2) | 3527160 | 0.000000000000001681011638091767100000000000000000 | -8.0 | -0.000000000000000672404655236706880000000000000000 | (12 ; 7 ; 1) | 1007760 | 0.000000000000000480289039454790560000000000000000 | -6.0 | -0.000000000000000144086711836437170000000000000000 | (12 ; 8 ; 0) | 125970 | 0.000000000000000060036129931848819000000000000000 | -4.0 | -0.000000000000000012007225986369764000000000000000 | (13 ; 0 ; 7) | 77520 | 0.000000000000000002052517262627310100000000000000 | -20.0 | -0.000000000000000002052517262627310100000000000000 | (13 ; 1 ; 6) | 542640 | 0.000000000000000014367620838391170000000000000000 | -18.0 | -0.000000000000000012930858754552053000000000000000 | (13 ; 2 ; 5) | 1627920 | 0.000000000000000043102862515173520000000000000000 | -16.0 | -0.000000000000000034482290012138815000000000000000 | (13 ; 3 ; 4) | 2713200 | 0.000000000000000071838104191955867000000000000000 | -14.0 | -0.000000000000000050286672934369107000000000000000 | (13 ; 4 ; 3) | 2713200 | 0.000000000000000071838104191955855000000000000000 | -12.0 | -0.000000000000000043102862515173514000000000000000 | (13 ; 5 ; 2) | 1627920 | 0.000000000000000043102862515173514000000000000000 | -10.0 | -0.000000000000000021551431257586757000000000000000 | (13 ; 6 ; 1) | 542640 | 0.000000000000000014367620838391173000000000000000 | -8.0 | -0.000000000000000005747048335356469700000000000000 | (13 ; 7 ; 0) | 77520 | 0.000000000000000002052517262627310100000000000000 | -6.0 | -0.000000000000000000615755178788193070000000000000 | (14 ; 0 ; 6) | 38760 | 0.000000000000000000057014368406314167000000000000 | -20.0 | -0.000000000000000000057014368406314167000000000000 | (14 ; 1 ; 5) | 232560 | 0.000000000000000000342086210437885050000000000000 | -18.0 | -0.000000000000000000307877589394096530000000000000 | (14 ; 2 ; 4) | 581400 | 0.000000000000000000855215526094712480000000000000 | -16.0 | -0.000000000000000000684172420875770000000000000000 | (14 ; 3 ; 3) | 775200 | 0.000000000000000001140287368126283400000000000000 | -14.0 | -0.000000000000000000798201157688398380000000000000 | (14 ; 4 ; 2) | 581400 | 0.000000000000000000855215526094712380000000000000 | -12.0 | -0.000000000000000000513129315656827430000000000000 | (14 ; 5 ; 1) | 232560 | 0.000000000000000000342086210437885000000000000000 | -10.0 | -0.000000000000000000171043105218942500000000000000 | (14 ; 6 ; 0) | 38760 | 0.000000000000000000057014368406314167000000000000 | -8.0 | -0.000000000000000000022805747362525667000000000000 | (15 ; 0 ; 5) | 15504 | 0.000000000000000000001266985964584759500000000000 | -20.0 | -0.000000000000000000001266985964584759500000000000 | (15 ; 1 ; 4) | 77520 | 0.000000000000000000006334929822923797200000000000 | -18.0 | -0.000000000000000000005701436840631417600000000000 | (15 ; 2 ; 3) | 155040 | 0.000000000000000000012669859645847596000000000000 | -16.0 | -0.000000000000000000010135887716678077000000000000 | (15 ; 3 ; 2) | 155040 | 0.000000000000000000012669859645847594000000000000 | -14.0 | -0.000000000000000000008868901752093316600000000000 | (15 ; 4 ; 1) | 77520 | 0.000000000000000000006334929822923798000000000000 | -12.0 | -0.000000000000000000003800957893754278600000000000 | (15 ; 5 ; 0) | 15504 | 0.000000000000000000001266985964584759500000000000 | -10.0 | -0.000000000000000000000633492982292379740000000000 | (16 ; 0 ; 4) | 4845 | 0.000000000000000000000021996284107374294000000000 | -20.0 | -0.000000000000000000000021996284107374294000000000 | (16 ; 1 ; 3) | 19380 | 0.000000000000000000000087985136429497177000000000 | -18.0 | -0.000000000000000000000079186622786547456000000000 | (16 ; 2 ; 2) | 29070 | 0.000000000000000000000131977704644245780000000000 | -16.0 | -0.000000000000000000000105582163715396630000000000 | (16 ; 3 ; 1) | 19380 | 0.000000000000000000000087985136429497177000000000 | -14.0 | -0.000000000000000000000061589595500648025000000000 | (16 ; 4 ; 0) | 4845 | 0.000000000000000000000021996284107374294000000000 | -12.0 | -0.000000000000000000000013197770464424576000000000 | (17 ; 0 ; 3) | 1140 | 0.000000000000000000000000287533125586592110000000 | -20.0 | -0.000000000000000000000000287533125586592110000000 | (17 ; 1 ; 2) | 3420 | 0.000000000000000000000000862599376759776370000000 | -18.0 | -0.000000000000000000000000776339439083798720000000 | (17 ; 2 ; 1) | 3420 | 0.000000000000000000000000862599376759776370000000 | -16.0 | -0.000000000000000000000000690079501407821060000000 | (17 ; 3 ; 0) | 1140 | 0.000000000000000000000000287533125586592110000000 | -14.0 | -0.000000000000000000000000201273187910614480000000 | (18 ; 0 ; 2) | 190 | 0.000000000000000000000000002662343755431408000000 | -20.0 | -0.000000000000000000000000002662343755431408000000 | (18 ; 1 ; 1) | 380 | 0.000000000000000000000000005324687510862816000000 | -18.0 | -0.000000000000000000000000004792218759776534400000 | (18 ; 2 ; 0) | 190 | 0.000000000000000000000000002662343755431408000000 | -16.0 | -0.000000000000000000000000002129875004345126400000 | (19 ; 0 ; 1) | 20 | 0.000000000000000000000000000015569261727669054000 | -20.0 | -0.000000000000000000000000000015569261727669054000 | (19 ; 1 ; 0) | 20 | 0.000000000000000000000000000015569261727669054000 | -18.0 | -0.000000000000000000000000000014012335554902147000 | (20 ; 0 ; 0) | 1 | 0.000000000000000000000000000000043247949243525151 | -20.0 | -0.000000000000000000000000000000043247949243525151 | ============== | ============ | ================================================== | ===== | =================================================== | Probabilité du gain : +0.365026235224989000 Probabilité du nul : +0.174914225073299400 Probabilité de la perte : +0.460059539701712590 Cumul des Probabilités : +1.000000000000001100 Espérance mathématique : -0.027027027027027070
D'ailleurs, c'est facile à les distinguer. Jeu de Pile ou Face : l'espérance est nulle.
La roulette européenne : l'espérance est de -0,027027.
Donc maintenant, tu peux comparer le résultat de :
--> la roulette européenne :+0.36502623.
--> la roulette française : +0.43810194.
--> le jeu de pile ou face : +0,44104418.
Et tu constates que la roulette française est moins intéressant que le jeu de pile ou face, mais bien meilleur que la roulette européenne.
Pourquoi ? Car on a les mêmes bases de comparaison !
En fait, tous ces calculs ne servaient strictement à rien, car l'espérance mathématique nous donnait déjà la réponse.
Sur vingt coups, l'espérance mathématique ramené à un coup, nous donne un jeu défavorable à la roulette, tandis que le jeu est équitable à Pile ou face. Et il n'existe pas de stratégies pouvant modifier ce résultat sur le long terme !
@+
19/12/2012 4:00 am
Par exemple, au jeu de pile ou face, si le zéro sort, le bilan est zéro !
Au jeu de Pile ou face le zéro sort ??
le jeu de pile ou face : +0,44104418
Quand on fait un cycle de 20 lancers avec une pièce, la probabilité d'avoir un cycle gagnant n'est pas 0.4414418.
Nous l'avons calculé auparavant, c'est 0.411901474
19/12/2012 5:28 am
Tu utilises les probabilités de la roulette, et il y en a trois. Faut-il que je te les redonne ou pas ?
Ensuite, qu'est-ce que tu ne comprends pas dans cette phrase :
Par exemple, au jeu de pile ou face, si le zéro sort, le bilan est zéro !
Tu attribues un zéro à la sortie du zéro au lieu d'un -1 comme sur la roulette européenne ou un -0,5 comme à la roulette française.
Le mot "bilan", c'est bien toi qui l'utilise dans ton classeur Excel, non ?
Et donc, avec ce nouveau bilan, tu simules un jeu de Pile ou Face.
Et comme par miracle, l'espérance mathématique est nulle !
Il te le faut ou pas ce nouveau tableau sur le jeu de pile ou face ? Car j'ai l'impression que tu ne comprends pas ce que je dis !
Et à ce jeu, tu obtiens bien +0.441044184688224540 de probabilité pour le gain.
Et je te le rappelle, ce calcul se fait avec la loi multinomiale et non la loi binomiale !
Ça y est tu es en phase maintenant ?
Si tu fait de même avec la roulette européenne, tu trouves (toujours pour la probabilité des gains) : +0.365026235224989000
Donc si compares ces trois résultats, tu trouves :
--> jeu de Pile ou Face : +0.441044184688224540
--> Roulette française : +0.438101948114484940
--> Roulette européenne : +0.365026235224989000
Donc la comparaison est possible, car tu as la même base de calcul !
@+
19/12/2012 5:45 am
La question est simple:
On joue à Pile ou Face et on parie systématiquement sur Pile
Quand on fait un cycle de 20 lancers, quelle est d'après toi la probabilité d'avoir un cycle gagnant ?
19/12/2012 8:22 am
Je viens de te répondre. Tu pourrais faire l'effort de me lire.
Avec la loi multinomiale, la réponse est : +0.441044184688224540.
C'est comme si tu avais une roulette sans zéro !
Maintenant, si tu considères la loi binomiale, la réponse est : +0.411901473999023438.
C'est comme le lancer d'une pièce au jeu de Pile ou Face.
Dans les deux cas, l'espérance mathématique est nulle !
@+
19/12/2012 12:52 pm
Je viens de te répondre. Tu pourrais faire l'effort de me lire.
Avec la loi multinomiale, la réponse est : +0.441044184688224540.
C'est comme si tu avais une roulette sans zéro !
Mais le jeu de Pile ou Face n'est pas une roulette.
Tu peux obtenir Pile ou Face mais pas zéro.
Maintenant, si tu considères la loi binomiale, la réponse est : +0.411901473999023438.
C'est la bonne probabilité.
Il n'y a pas 2 probabilités !
Quand tu auras fait 10000 cycles de 20 lancers, la moyenne des cycles gagnants ne peut pas être à la fois 4119 et 4410 !
C'est soit l'un soit l'autre. C'est l'un.
1 ; 11 ; 8 | 1511640 | 0.046301119416361385000000000000000000000000000000
Voici un détail de tes calculs.
Selon toi, on a une probabilité de 4.63% de faire 11 Piles, 8 Faces et 1 zéro en lançant 20 fois une pièce ???
Tu te sens bien ?
