A die sets the range for your next pick

Question

Roll a fair six-sided die and call the result N. Then pick a whole number K uniformly from \{1, 2, \dots, N\}. What is the expected value of K?

Accepted Answer

The pick K depends on the roll N, so condition on N and apply the tower rule: E[K] = E\big[\,E[K \mid N]\,\big]. Step 1, freeze the roll. Given N, the pick K is uniform on \{1, \dots, N\}, and a uniform pick on \{1, \dots, m\} averages \frac{m+1}{2}. So E[K \mid N] = \frac{N + 1}{2}. Step 2, average over the roll. The die averages E[N] = \frac{1+2+3+4+5+6}{6} = 3.5. Then E[K] = E\!\left[\frac{N + 1}{2}\right] = \frac{E[N] + 1}{2} = \frac{3.5 + 1}{2} = 2.25. So E[K] = \frac{9}{4} = 2.25. Takeaway. When one draw sets the range of the next, condition on the first, use the uniform mean \frac{m+1}…

A die sets the range for your next pick

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