Pig is a dice game in which a player repeatedly chooses whether to bank a turn's score or risk losing it by rolling again. In the version considered here, two players take turns using a fair six-sided die, and the first to bank at least 100 points wins. Successive rolls are independent.
A turn starts with a total of 0. Rolling 2, 3, 4, 5, or 6 adds that number to the turn total. Rolling 1 loses the entire turn total and ends the turn, without changing previously banked points. Holding adds the turn total to the player's banked score and ends the turn (Neller and Presser 2004, p. 25).
For a comparison that ignores the winning threshold, suppose the current turn total is 
. Rolling once more and then holding whenever
the roll is not 1 changes the banked score, relative to holding immediately, by 
with probability 
and by each of 2, 3, 4, 5, and 6 with
probability 
. The expectation value
of this change is therefore
 |
Thus another roll is favorable in this comparison when 
, unfavorable when 
, and neutral when 
.
This calculation motivates the rule of holding at a turn total of at least 20, but maximizing expected points in a turn is different from maximizing the probability of winning the game. Optimal play depends on both players' banked scores as well as the current turn total (Neller and Presser 2004, pp. 26-27).
