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.)


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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