Wikipedia:Reference desk/Archives/Mathematics/2018 January 28

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January 28

What is the probability of winning a round in this game?

Chicago dice [1] - Each round is an independent event; the probability of winning each round is 1/36, 2/36, etc.; totaling each probability gives 36/36 = 1, clearly a wrong answer since it's quite conceivable that every round sees rotten luck. Where did I go wrong? Thank you. Imagine Reason (talk) 14:12, 28 January 2018 (UTC)

What are you looking for exactly? Do you mean the probability of winning at least one round? That would be

$1-[(1-1/36)(1-2/36)\cdots (1-1/36)]=0.6541\ldots$

–Deacon Vorbis (carbon • videos) 14:26, 28 January 2018 (UTC)

In this problem, and many others, it is easier to calculate the probability of not winning and subtract it from 1.0, as in the answer above. Bubba73 ^{You talkin' to me?} 19:15, 28 January 2018 (UTC)

You didn't go wrong! You successfully calculated the expected value of the number of rounds won in one game. By realizing that the rounds were independent events, you calculated the expected value the easy way. The harder way would have been to calculate p_n, the probability of wining exactly n rounds in a single game, and then sum $\operatorname {E} [n]=\sum _{n=0}^{11}n\cdot p_{n}$, getting the same value 1. -- ToE 20:35, 28 January 2018 (UTC)

Probability pn of wining exactly n rounds in a single game for n = 0..11. -- ToE 21:10, 28 January 2018 (UTC)

$n$ $p_{n}$ $n\cdot p_{n}$ partial sum $\operatorname {\sum } _{i=0}^{n}i\cdot p_{i}$ 0 0.34589070627536 0 0 1 0.390575891815729 0.390575891815729 0.390575891815729 2 0.194858582093977 0.389717164187955 0.780293056003683 3 0.0565944763062653 0.169783428918796 0.950076484922479 4 0.0106104904557046 0.0424419618228185 0.992518446745298 5 0.00134517629161697 0.00672588145808485 0.999244328203382 6 0.00011736207591064 0.000704172455463842 0.999948500658846 7 7.02554423422569e-06 4.91788096395798e-05 0.999997679468486 8 2.81845256132349e-07 2.25476204905879e-06 0.999999934230535 9 7.19065555582081e-09 6.47159000023873e-08 0.999999998946435 10 1.04634415130382e-10 1.04634415130382e-09 0.999999999992779 11 6.56426694669901e-13 7.22069364136891e-12 1 = Expected value

Actually, the expectation is additive even if the events are not assumed to be independent. -- Meni Rosenfeld (talk) 22:31, 29 January 2018 (UTC)

Further study: If you play your Chicago Dice with only 1 die, giving six rounds per game numbered 1..6, then you have a 1/6 chance of winning each round and the expected number of rounds won per game is still 1. What if you play with 3 dice, giving 16 rounds, numbered 3..18? What if you play with n dice, giving 5n+1 rounds, numbered n..6n? What are the expected number of rounds won per game and why? -- ToE 23:41, 29 January 2018 (UTC)

A relevant sequence is OEIS: A063260, Sextinomial (also called hexanomial) coefficient array, which "can be used to calculate the number of occurrences of a given roll of n six-sided dice". -- ToE 23:52, 29 January 2018 (UTC)