Eight coins with one optional reflip each

Question

Eight fair coins are tossed together. After seeing the result you may choose any subset of the coins and toss each of those once more (each coin can be retossed at most once). You then receive $1 for every coin showing heads. If you play optimally, what is the game worth?

Short answer

$6. Keep every head and retoss every tail. A coin then ends on tails only if both of its tosses are tails, probability $\tfrac14$ , so each coin is worth $\tfrac34$ and eight coins give $8\cdot\tfrac34=6$ by linearity of expectation.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Optimisation Problems

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