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## Rolling the Dice

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #26344 Fun: (2.83) Difficulty: (2.09) Category: Probability Submitted By: kiss Corrected By: Shadows

A gambler goes to bet. The dealer has 3 dice, which are fair, meaning that the chance that each face shows up is exactly 1/6.

The dealer says: "You can choose your bet on a number, any number from 1 to 6. Then I'll roll the 3 dice. If none show the number you bet, you'll lose \$1. If one shows the number you bet, you'll win \$1. If two or three dice show the number you bet, you'll win \$3 or \$5, respectively."

Is it a fair game?

It's a fair game. If there are 6 gamblers, each of whom bet on a different number, the dealer will neither win nor lose on each deal.

If he rolls 3 different numbers, e.g. 1, 2, 3, the three gamblers who bet 1, 2, 3 each wins \$1 while the three gamblers who bet 4, 5, 6 each loses \$1.

If two of the dice he rolls show the same number, e.g. 1, 1, 2, the gambler who bet 1 wins \$3, the gambler who bet 2 wins \$1, and the other 4 gamblers each loses \$1.

If all 3 dice show the same number, e.g. 1, 1, 1, the gambler who bet 1 wins \$5, and the other 5 gamblers each loses \$1.

In each case, the dealer neither wins nor loses. Hence it's a fair game.

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