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

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[X^{2}] = P(sum 7)($4)^{2} + P(sum
11)($4)^{2} = 7.56

Variance = 7.56 - 1.89^{2} = 3.99 ~ 4

Standard deviation = **2**

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