Question

You pay $1 to play a game. The game consists of rolling a pair of dice. If you observe a sum of 7 or 11 you receive $4. If not, you receive nothing. Compute the expected value and standard deviation for this game?

Answer 1

Total outcomes of rolling a pair of dice = 36
Number of outcomes in which sum is 7 = #{(3,4);(4,3);(1,6);(6,1);(5,2);(2,5)} = 6
P(sum 7) = 6/36 = 1/6
Number of outcomes in which sum is 11 = #{(5,6);(6,5)} = 2
P(sum is 11) = 11/36
Expected value of the game = P(sum 7)($4) + P(sum 11)($4) = (1/6 x 4) + (11/36 x 4) = 1.89
Expected value of the game E[X] = 0.89
E[X²] = P(sum 7)($4)² + P(sum 11)($4)² = 7.56
Variance = 7.56 - 1.89² = 3.99 ~ 4
Standard deviation = 2