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?
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?
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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
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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Yes, it is exactly fair.
Probability of getting my number on 3 of 3 dice: 0.005
Probability of getting my number on 2 of 3 dice: 0.069
Probability of getting my number on 1 of 3 dice: 0.347
Probability of getting my number on 0 of 3 dice: 0.579
This means in 1000 plays, I would win $5 5 times; I would win $3 69 times; I would win $1 347 times and I would lose $1 579 times. Add all the winnings and subtract the losses, and you are left with $0.00
Probability of getting my number on 3 of 3 dice: 0.005
Probability of getting my number on 2 of 3 dice: 0.069
Probability of getting my number on 1 of 3 dice: 0.347
Probability of getting my number on 0 of 3 dice: 0.579
This means in 1000 plays, I would win $5 5 times; I would win $3 69 times; I would win $1 347 times and I would lose $1 579 times. Add all the winnings and subtract the losses, and you are left with $0.00