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Paul
Posts:
764
Registered:
7/12/10


Re: probability question about the dice game
Posted:
Feb 14, 2013 4:29 PM


On Thursday, February 14, 2013 5:13:26 PM UTC, duncan smith wrote: > On 14/02/13 16:17, pepstein5@gmail.com wrote: > > > On Thursday, February 14, 2013 3:09:37 PM UTC, Jussi Piitulainen wrote: > > >> David C. Ullrich writes: > > >> > > >> > > >> > > >>> On 14 Feb 2013 15:50:27 +0200, Jussi Piitulainen wrote: > > >> > > >>> > > >> > > >>>> starwayinc@gmail.com writes: > > >> > > >>>> > > >> > > >>>>> two players Ann and Bob roll the dice. each rolls twice, Ann wins if > > >> > > >>>>> her higher score of the two rolls is higher than Bobs, other wise > > >> > > >>>>> Bob wins. please give the analyse about what is the probability that > > >> > > >>>>> Ann will win the game > > >> > > >>>>> > > >> > > >>>> > > >> > > >>>> Because > > >> > > >>>> > > >> > > >>>> >>> from itertools import product > > >> > > >>>> >>> die = {1,2,3,4,5,6} > > >> > > >>>> >>> dice = set(product(die, die, die, die)) > > >> > > >>>> >>> sum(int(max(a,b) > max(c,d)) for a,b,c,d in dice) > > >> > > >>>> 505 > > >> > > >>>> >>> sum(int(max(a,b) <= max(c,d)) for a,b,c,d in dice) > > >> > > >>>> 791 > > >> > > >>> > > >> > > >>> Cool. > > >> > > >>> > > >> > > >>>> I'd say her odds are 505 for and 791 against. I hope my gambling > > >> > > >>>> vocabulary is not too far off. > > >> > > >>> > > >> > > >>> The terminology would be "her odds of winning are 505 to 791". > > >> > > >> > > >> > > >> Thanks, that looks familiar. I'm sure I've also seen "for" (or maybe > > >> > > >> "on") and "against" _somewhere_ in such expressions. > > > > > > David Ullrich is wrong. "X to Y" means that the probability of winning is > > > (X + Y)/Y. > > > > Greater than 1? > > > > > "X to Y against" means that the probability of winning is (X + Y)/Y and X is larger than Y. > > > "X to Y on" means that the probability of winning is (X + Y)/Y and X is less than Y. > > > > > > In this context "on" and "against" are redundant. However, these words enable useful abbreviations as follows. "Twos on" means " 1 to 2 " "Twos against" means "2 to 1". You can also write a slash "/" instead of the word "to". > > > > > > > No. Odds of X to Y (or X to Y on) means a probability of X/(X+Y). Odds > > of X to Y against means means a probability of Y/(X+Y) or 1  X/(X+Y). >
No! This is wrong for at least 3 reasons. (My initial post was wrong because I gave the reciprocals of the probabilities, but my second post was correct.) Reason 1: you say "(or X to Y on)." This implies that "Odds of X to Y" is synonymous with "Odds of X to Y on". However, they are not synonymous at all. A bet that is "on" means that you are betting on something that is considered an event with probability > 50%, whereas "Odds of X to Y" has no such restriction. Reason 2: "Odds of X to Y" means that if you win your bet after staking Y, you get paid X and also get your stake of Y refunded. If it was a fair bet (a simplification of course, because bookies expect to make money), the odds would therefore be Y/(X + Y). Reason 3: If someone doesn't use the words "on" or "against", the default convention is "against". You wrongly state that the default convention is "on".
Paul Epstein



