A game on the sum of two dice

Topic: Random variables and distributions · more questions on this topic
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Two fair dice are thrown and Z is the sum of the faces.

1. Give the probability mass function of Z.

2. Calculate \operatorname{Var}(Z).

3. A player is paid (Z-7)^2 and pays 5 to play. Find the expected profit from 100 independent plays.

Worked answer (click to show)

1. Of the 36 equally likely ordered pairs, 6-|z-7| have sum z, so

\mathbb{P}(Z=z)=\frac{6-|z-7|}{36},\qquad z=2,3,\ldots,12.

2. Z is the sum of two independent faces, each with mean 7/2 and variance 35/12, so \mathbb{E} Z=7 and \operatorname{Var}(Z)=2\times35/12=35/6\approx5.83.

3. The payout is (Z-7)^2, whose expectation is \operatorname{Var}(Z)=35/6. The expected profit per play is 35/6-5=5/6, so over 100 independent plays it is 100\times5/6\approx83.3.