Pay $5 to roll, win ten dollars minus the face

Question

You pay 5 to play. You roll one fair six-sided die and win (10 minus the face value). So a 1 pays 9, a 4 pays 6, and a 6 pays 4. Low rolls are better here. On average, how much do you win or lose per play, after the fee?

Accepted Answer

Find the average payout, then subtract the fee. The payout is 10 minus the face value. A handy shortcut: the average of "10 minus the die" is just 10 minus the average of the die. The average face value is: \frac{1+2+3+4+5+6}{6} = 3.5. So the average payout is: 10 - 3.5 = 6.5 	ext{ dollars.} If you doubt the shortcut, you can list the six payouts (9, 8, 7, 6, 5, 4) and average them: their sum is 39, and 39/6 = 6.5. Same answer. Now subtract the 5 fee: EV = 6.5 - 5 = 1.5. You gain an average of 1.50 per play. Takeaway. The average of "a constant minus the die" is that constant minus 3.5. Flip…

Pay $5 to roll, win ten dollars minus the face

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