A coin flip decides whether you roll one die or two

Question

You flip a fair coin. On heads you roll one fair die; on tails you roll two fair dice. Your score is the total of the pips shown. What total should you expect?

Accepted Answer

Two branches set by the coin. The law of total expectation (tower rule) averages the two branch outcomes, weighted by the coin: E[	ext{total}] = E\big[\,E[	ext{total} \mid 	ext{coin}]\,\big]. Step 1, score each branch. A fair die averages 3.5. - Heads (one die): E[	ext{total} \mid 	ext{heads}] = 3.5. - Tails (two dice): E[	ext{total} \mid 	ext{tails}] = 3.5 + 3.5 = 7. Step 2, weight by the fair coin, 	frac12 each: E[	ext{total}] = 0.5 	imes 3.5 + 0.5 	imes 7 = 1.75 + 3.5 = 5.25. You should expect a total of 5.25, that is \frac{21}{4}. Takeaway. Even when the branches involve differ…

A coin flip decides whether you roll one die or two

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