Pay $4 to roll a die for its face value

Question

You pay 4 to play. You roll one fair six-sided die and receive its face value in dollars (a 3 pays 3, a 6 pays 6, and so on). On average, how much do you win or lose per play, after the fee?

Accepted Answer

First find the average payout from the die, then subtract the cost to play. A fair die shows 1, 2, 3, 4, 5, or 6, each equally likely. The average of these six numbers is their sum divided by six: \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5. So on average the die hands you 3.50. But you paid 4 to play, and that fee comes out every single time. Net average per play: EV = 3.5 - 4 = -0.5. You lose an average of 50 cents each play. The prize sounds fun, but the 4 entry fee is larger than the 3.50 the die is worth, so the game quietly bleeds you over time. Takeaway. For a die that pays its face val…

Pay $4 to roll a die for its face value

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