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

Mathematics desk
< January 27 << Dec | January | Feb >> Current desk >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


January 28

edit

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

edit

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)[reply]

What are you looking for exactly? Do you mean the probability of winning at least one round? That would be
 
Deacon Vorbis (carbon • videos) 14:26, 28 January 2018 (UTC)[reply]
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)[reply]
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 pn, the probability of wining exactly n rounds in a single game, and then sum  , getting the same value 1. -- ToE 20:35, 28 January 2018 (UTC)[reply]
Probability pn of wining exactly n rounds in a single game for n = 0..11. -- ToE 21:10, 28 January 2018 (UTC)[reply]
      partial sum  
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)[reply]
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)[reply]
A relevant sequence is OEISA063260, 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)[reply]