Wikipedia:Reference desk/Archives/Mathematics/2021 March 25

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March 25

Strange dice game

Two players play a dice game. They both use the same die. The game goes like this, you roll the die until you get the same number twice in a row. Your score is the number of rolls. The player with the highest number of roll is the winner. If tie then repeat the game again.

Example:

Player 1: 4,3,6,1,2,2 (score 6 rolls)

Player 2: 1,6,2,5,4,1,3,3 (score 8 rolls)

Player 2 is the winner

Question: Does it matter if the die is a "fair" dice or a "bias" dice. 1.159.120.235 (talk) 01:55, 25 March 2021 (UTC)

No. Unless one player knows which way the bias is. See https://www.popularmechanics.com/technology/a22856/dice-mathematically-fair/ Secondly, is it even possible to manufacture a fair die? How would you test it? Any test by throwing the die thousands/millions of times would result in the die being worn down. This would introduce further bias. Obviously casinos and the like use well manufactured dice that have no discernible bias. 41.165.67.114 (talk) 06:43, 25 March 2021 (UTC)

The scores obtained by the players are outcomes of a random variable that has a certain probability distribution. In the game as described, each throw of a die after the first throw of a play constitutes a Bernoulli trial (where "success" means "same number as eyes as the previous throw"), so (for a fair six-sided die) the scores, after subtracting $1$ for the first throw, have the geometric distribution with parameter $p={\tfrac {1}{6}}$. However, this is irrelevant to the question. It is completely irrelevant which specific discrete probability distribution this is out of a myriad of possibilities. What matters, is solely that it is the same for both players — in statistical jargon, that their random variables are independent and identically distributed. Under that assumption, their chances of winning are the same. It does not matter whether they play with fair or biased dice, nor whether one player has more information about the distribution, since they cannot use this knowledge to influence their outcomes.  --Lambiam 08:33, 25 March 2021 (UTC)