Random number of dice, then their total

Question

You roll one fair six-sided die and call the result N. You then roll N more fair six-sided dice and add up all the faces they show. What is the expected value of that total?

Accepted Answer

The total has two layers of randomness stacked on top of each other: first how many dice you roll (N), then what those dice show. When randomness comes in layers, the clean tool is the law of total expectation, also called the tower rule: E[	ext{total}] = E\big[\,E[	ext{total} \mid N]\,\big]. Read that right to left. The inner piece E[	ext{total} \mid N] pretends you already know N, which turns a hard problem into an easy one. The outer E[\cdot] then averages that easy answer back over the randomness in N. Step 1, freeze the first layer. Suppose you already know N = n. Then you are simply …

Random number of dice, then their total

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