Pay $8 to roll, win double the face plus one dollar

Question

You pay 8 to play. You roll one fair six-sided die and win twice its face value plus a flat 1 bonus. So a 3 pays 7, a 6 pays 13. 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 has two parts: twice the face value, plus a fixed 1. Handle each part. The average face value of a fair die is 3.5. Twice that is: 2 	imes 3.5 = 7. Then add the flat 1 bonus, which is paid every time regardless of the roll, so it just adds 1 to the average: 7 + 1 = 8 	ext{ dollars average payout.} Now subtract the 8 fee: EV = 8 - 8 = 0. The game breaks even on average. Over many plays you neither gain nor lose, since the 8 you pay exactly matches the 8 the die is worth. Takeaway. A flat bonus adds its full amount to the average, and…

Pay $8 to roll, win double the face plus one dollar

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