Pay $3 to roll a die for its face value

Question

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

Accepted Answer

Find the average payout of the die, then subtract the entry fee. A fair die shows 1 through 6 equally often, and the average of those numbers is: \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5. So the die pays 3.50 on average. This time the fee is only 3, so: EV = 3.5 - 3 = 0.5. You gain an average of 50 cents per play. The lower fee flips the game into your favor. Compared to the same die with a 4 fee, this single dollar of price difference is the whole story: it turns a losing game into a winning one. Takeaway. The die is worth 3.50 on average. Any fee below that gives you a positive edge, and …

Pay $3 to roll a die for its face value

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