### Brain Teasers

# Balls in a Jar

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

You're walking down a street, when you see some people gambling on the roadside.

The dealer says:

"There are 10 black balls and 10 white balls in this jar. You are blindfolded and you randomly pick 10 balls. If 5 of the balls you picked are black and the other 5 are white, you lose $2. Otherwise you win $1."

Is the gamble fair?

The dealer says:

"There are 10 black balls and 10 white balls in this jar. You are blindfolded and you randomly pick 10 balls. If 5 of the balls you picked are black and the other 5 are white, you lose $2. Otherwise you win $1."

Is the gamble fair?

### Hint

You know that, in most gambles, the dealer wins more than he loses. If not, why does he hold the gamble?### Answer

The number of combinations to take 10 balls out of 20 is 20!/(10!)^2=184756.The number of combinations to take 5 balls of each color is (10!/(5!)^2)^2=63504.

This means in 184756 games you lose 63504 games and win 121252 games in average, totaling a loss of $5756. In other words, you lose about $0.03116 per game in average.

Therefore, the gamble is not fair.

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