Pay $5 to roll a die for double its face value

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You pay 5 to play. You roll one fair six-sided die and receive twice its face value in dollars (a 3 pays 6, a 6 pays 12). On average, how much do you win or lose per play, after the fee?

Accepted Answer

Work out the average payout, then subtract the fee. The average face value of a fair die is: \frac{1+2+3+4+5+6}{6} = 3.5. But here the payout is double the face value. Doubling every payout doubles the average payout too, so the die is worth: 2 	imes 3.5 = 7 	ext{ dollars on average.} Now subtract the 5 fee: EV = 7 - 5 = 2. You gain an average of 2 per play. The key move is that scaling every prize by a constant (here, times two) scales the average by that same constant, so you did not need to redo the whole calculation from scratch. Takeaway. Multiplying every payout by a constant multipli…

Pay $5 to roll a die for double its face value

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