Pick a coin from a mixed bag, then flip ten times

Question

A bag of coins is 40\% fair coins (heads probability 0.5) and 60\% trick coins (heads probability 0.8). You pull one coin at random and flip it 10 times. How many heads should you expect?

Accepted Answer

First the coin type is drawn, then 10 flips run at that coin's bias. Use the tower rule, conditioning on which coin you grabbed: E[	ext{heads}] = E\big[\,E[	ext{heads} \mid 	ext{coin}]\,\big]. Step 1, freeze the coin. Given a coin with bias p, ten flips average 10\,p heads. So the fair coin gives 10 	imes 0.5 = 5 and the trick coin gives 10 	imes 0.8 = 8. Step 2, average over the coin type. Weight those two conditional answers by how often each coin is drawn: E[	ext{heads}] = 0.40 	imes 5 + 0.60 	imes 8 = 2 + 4.8 = 6.8. You expect 6.8 heads. Equivalently, the average bias is E[p] = 0.…

Pick a coin from a mixed bag, then flip ten times

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