Hat Party

Probability Level pending

10 people enter a party, each with a different hat. They place their hats on a rack.

As they leave at the end of long evening, they are unable to tell their hats apart and each pick one at random. What is the probability that exactly 9 out of 10 picked their own hat?

9 ! 10 ! \frac{9!}{10!} 1 10 \frac{1}{10} 0 0 1 10 ! \frac{1}{10!}

Jason Dyer by Jason Dyer

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1 solution

Jason Dyer Staff
Oct 24, 2016

It is impossible for exactly 9 out of 10 hats to be correct. If 9 out of 10 hats are taken correctly, the 1 0 th 10^\text{th} hat has only one person it can go to - the correct head.

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Clever question. People who heard about the matching problem might want to solve this from the first principle, whereas you could easily solve this particular problem just by observing that only even number of mismatches is possible.

Samrat Mukhopadhyay - 4 years, 6 months ago

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Odd number of mismatches are also possible like we assume there are 3 people who have mismatched hat as 1st person can wear 2nd hat, 2nd wear 3rd hat and 3rd wear 1st hat.

Archit Agrawal - 4 years, 6 months ago

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Oh, yeah, correct, stupid me. Only the case of 1 1 mismatch is not possible.

Samrat Mukhopadhyay - 4 years, 6 months ago

I also thought it must be hard, until I thought about the impossibility to solve exactly 5 sides of a Rubik's cube.

Jingyang Tan - 4 years, 6 months ago

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