Fair price of a payout that doubles per head

MediumExpected Value Market Making & Betting Games~4m

A fair coin is flipped four times. The game pays 1 dollar, doubled once for every head that appears (no heads pays 1 dollar, two heads pay 4 dollars, four heads pay 16 dollars). What is the fair price of the game, in dollars to four places?

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Write the payout as $2^H$ and split it into one factor per flip: $2^H = \prod_i 2^{X_i}$ .
Each factor has mean $\tfrac32$ ; the flips are independent.

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5.0625 (± 0.001)

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Fair price of a payout that doubles per head

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