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Question

Jonathan plays a game where he first roll one dice, followed by two dices then lastly, three dices. If he gets a smaller sum every time, he wins the game. Find the probability he wins the game.

Answer 1

The smallest sum that can result from rolling 3 standard dice is 3.

The sample space then, in the form ( X,Y,Z) , where X is the sum of 3 dice,

Y is the sum of 2 dice, and Z is the result of 1 dice, with X<Y<Z , is

(3,4,5)

(3,4,6)

(3,5,6)

(4,5,6)

There are 36 possibilities for rolling 2 dice and 216 possibilities for rolling 3 dice.

Rolling 2 dice, outcome of 4 is 1 and 3, 2 and 2, 3 and 1, with probability 3/36

outcome of 5 is 1 and 4, 2 and 3, 3 and 2, 4 and 1, with probability 4/36

Rolling 3 dice, outcome of 3 is clearly 1 and 1 and 1 , with probability 1/216

outcome of 4 is 1 and 1 and 2 or 1 and 2 and 1 or 2 and 1 an 1, with probability 3/216

So here are the probabilities:

(3,4,5) has probability 1/216 * 3/36 *1/6 = 3/46656 = 1/15552

(3,4,6) has probability 1/216 * 3/36 * 1/6 = 1/15552

(3,5,6) has probability 1/216 * 4/36 *1/6 = 4/46656 = 1/11664

(4,5,6) has probability 3/216 * 4/36 * 1/6 = 12/46656 = 1/3888

The total probability is then (3+1+4+12)/46656 = 20/46656 = 10/ 23328 = 5/11664

which is not even 1/20 % = 0.05%