Average of the larger of two dice

Question

Two fair six-sided dice are rolled and you keep whichever shows the higher number. What is the expected value of the number you keep?

Short answer

$161/36\approx 4.47$ . $P(\max\le k)=k^2/36$ , so $P(\max=k)=\tfrac{2k-1}{36}$ . Then $E=\sum_k k\,\tfrac{2k-1}{36}=\tfrac{161}{36}$ .

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Lessons behind this question

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

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