You are approached by a compulsive gambler with the following proposal. You are to flip a fair coin four times. If heads and tails both appear twice each, he will pay you $11. If any other combination of heads and tails appears, you have to pay him only $10. Do you take the wager? Click below for the answer.

This is not a good gamble. Below are all the possible outcomes for four successive coin flips.

Notice that exactly two heads and two tails only appear six out of sixteen times, so you can only expect to win this game about 37.5% of the time. At the offered stakes ($11 for a win, $10 for a loss) you'd be losing an average of around $2.12 every time you play.

SPOILER ALERT: Don't read this comment unless you've already worked the problem and/or looked at the solution.

A tad confused here -- /on average/, if I do this 16 times, I will win 6x ($66 in my pocket) and lose 10x (pay out $100). So every 16 runs, I lose $34. Seems like on average I lose $34/16 =~ $2.12 each time? Maybe the $3.12 is a typo?

## 2 comments:

SPOILER ALERT: Don't read this comment unless you've already worked the problem and/or looked at the solution.

A tad confused here -- /on average/, if I do this 16 times, I will win 6x ($66 in my pocket) and lose 10x (pay out $100). So every 16 runs, I lose $34. Seems like on average I lose $34/16 =~ $2.12 each time? Maybe the $3.12 is a typo?

Always enjoy the puzzles.

Jimbo,

Yes, you're right! Thank you for the correction.

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