Is chuck-a-luck a fair game?
In the carnival game chuck-a-luck you bet 1 dollar on a number from 1 to 6, and three fair dice are rolled. If your number shows on exactly one die, you get your dollar back plus 1 dollar. On two dice, your dollar back plus 2. On all three, your dollar back plus 3. If your number does not appear, you lose your dollar.
Many players reason: "Three dice, six numbers, so my number should show up half the time, and when it does I get paid at least even money. Surely this is fair or better."
Compute the expected profit per dollar bet. Is the game fair?
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The chance that your number shows on at least one die is not one half. Count the exact probabilities of 0, 1, 2 and 3 matches.
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