Math Forum Discussions

Paul

Posts: 258
Registered: 7/12/10

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

Plain Text

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:
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> >>
>
> >>
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> >>
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> >>> On 14 Feb 2013 15:50:27 +0200, Jussi Piitulainen wrote:
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> >>
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> >>>
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> >>
>
> >>>> starwayinc@gmail.com writes:
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> >>
>
> >>>>
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> >>
>
> >>>>> two players Ann and Bob roll the dice. each rolls twice, Ann wins if
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> >>
>
> >>>>> her higher score of the two rolls is higher than Bobs, other wise
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> >>
>
> >>>>> Bob wins. please give the analyse about what is the probability that
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> >>
>
> >>>>> Ann will win the game
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> >>
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> >>>>>
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> >>
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> >>>>
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> >>
>
> >>>> Because
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> >>
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> >>>>
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> >>
>
> >>>> >>> from itertools import product
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> >>
>
> >>>> >>> die = {1,2,3,4,5,6}
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> >>
>
> >>>> >>> dice = set(product(die, die, die, die))
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> >>
>
> >>>> >>> sum(int(max(a,b) > max(c,d)) for a,b,c,d in dice)
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> >>
>
> >>>> 505
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> >>
>
> >>>> >>> sum(int(max(a,b) <= max(c,d)) for a,b,c,d in dice)
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> >>
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> >>>> 791
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> >>
>
> >>>
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> >>
>
> >>> Cool.
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> >>
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> >>>
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> >>
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> >>>> I'd say her odds are 505 for and 791 against. I hope my gambling
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> >>
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> >>>> vocabulary is not too far off.
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> >>
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> >>>
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> >>
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> >>> The terminology would be "her odds of winning are 505 to 791".
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> >>
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> >>
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> >>
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> >> Thanks, that looks familiar. I'm sure I've also seen "for" (or maybe
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> >>
>
> >> "on") and "against" _somewhere_ in such expressions.
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> >
>
> > David Ullrich is wrong. "X to Y" means that the probability of winning is
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> > (X + Y)/Y.
>
>
>
> Greater than 1?
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>
>
> > "X to Y against" means that the probability of winning is (X + Y)/Y and X is larger than Y.
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> > "X to Y on" means that the probability of winning is (X + Y)/Y and X is less than Y.
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> >
>
> > 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".
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> >
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>
>
> No. Odds of X to Y (or X to Y on) means a probability of X/(X+Y). Odds
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> 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