Second die whose size depends on the first roll

Question

Roll a standard six-sided die and call the result $k$ . Then roll a fair die with $6+k$ faces labelled $1$ to $6+k$ . What is the expected value of the second roll?

Short answer

$21/4$ . Given $k$ , the second roll averages $(7+k)/2$ . By the tower property, the overall mean is $\tfrac{7+3.5}{2}=\tfrac{21}{4}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Counting and Combinatorics
Conditional Expectation and Variance Revisited

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