A programming student was recently interested in simulating and analyzing the dice game Pig. While working out some of the relevant probabilities and expected values for the original game, I came up with the following twist, that I think is even simpler to describe, but still presents some interesting complexity in game play and analysis:
As in Pig, you and another player each take a turn rolling a single die, repeatedly as many times as you wish, until either:
- You decide to stop, at which point your score is the sum of all of your rolls in the turn; or
- You roll a 1, at which point your turn ends with a score of zero (i.e., you lose all of your rolls in the turn).
Rather than alternating turns with the winner being the first to reach 100 points, in this variant you each take just a single turn, simultaneously and separately, i.e., without knowledge of the other player’s outcome. The highest score wins.
What is your optimal strategy for playing this game? And for any given strategy, what is the probability distribution of outcomes (turn totals)?