Three coins, a dollar a head, and a matching bonus

Question

You flip three fair coins. You collect \1 for each head that appears. On top of that, if all three coins come up the same (all heads or all tails), you receive a \5 bonus. What is the expected value of one play?

Accepted Answer

Split the payout into two parts and add their expected values. Heads part. Each coin contributes \1 with probability 	frac12, so the expected number of heads, and the expected dollars from heads, is 3 	imes 	frac{1}{2} = 1.5. Bonus part. All three match only for HHH or TTT, which is 2 of the 8 equally likely outcomes: P(	ext{all same}) = \frac{2}{8} = \frac{1}{4}, \qquad E[	ext{bonus}] = \frac{1}{4}	imes 5 = 1.25. Add the pieces: E[	ext{play}] = 1.5 + 1.25 = 2.75. The game is worth \2.75. Takeaway. Expected value adds even when the parts are dependent (the bonus depends on the same fli…

Three coins, a dollar a head, and a matching bonus

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