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I would like to know if my following reasoning is correct.

The problem

A casino sells coupons to buy into exactly one game for $10. All winnings, if any, are kept by the purchaser. The casino takes back the coupon regardless of a win or loss.

What is a reasonable price to pay for the coupon?

My working

Say the gambler is willing to pay a price $P$ for the coupon.

A win yields a profit of $(10 - P)$ while a loss gives $-P$ in profit, the price of buying the coupon.

Consider the long run situation where $m$ wins and $n$ games are won and lost. Then the expected winnings are $$(10 - P) m + (-P) n = 10m - P(m + n)$$ The gambler wants a profit i.e. for expected winnings to be positive, so $$10m - P(m + n) > 0 \leftrightarrow P < 10 \displaystyle \frac{m}{m + n}$$

but $\displaystyle \frac{m}{m + n}$ represents the theoretical win probability, so $P$ is any price less than 10 times the win probability on the chosen game (since the game is only played once.)

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This is correct. You should be willing to pay the expected value of the game, which is $\$10$ times the winning probability given that you do not lose anything except the coupon price.

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