Balls in a JarProbability 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?
HintYou know that, in most gambles, the dealer wins more than he loses. If not, why does he hold the gamble?
AnswerThe 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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