Trading maths round.
The larger of two dice
The question
Two fair six-sided dice are rolled. What is the expected value of the larger?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
161/36, about 4.47.
Use the tail formula: the expectation of a non-negative integer variable is the sum over k of P(X >= k).
P(max >= k) = 1 - ((k-1)/6)^2. Summing for k from 1 to 6 gives 1 + 35/36 + 32/36 + 27/36 + 20/36 + 11/36.
That total is 161/36, roughly 4.47.
Counting cases works too but is slower out loud, and the tail formula generalises immediately to n dice or to the minimum.
What it is testing
Knowing the tail-sum identity and choosing it over brute force.
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