Solving the Matching Hats Prob: N Men, No Hats Selected & k Hats Selected

  • Thread starter eku_girl83
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In summary, this conversation discusses the probability of men randomly selecting their own hats after throwing them into a room and mixing them up. The first question asks for the probability that none of the men select their own hats, while the second question asks for the probability that exactly k of the men select their own hats. The conversation suggests using numbers to better understand the problem and mentions the inclusion exclusion principle as a possible approach.
  • #1
eku_girl83
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Suppose that each of N men at a party throws his hat into the center of the room. The hats are first mixed up, and then each man randomly selects a hat. What is the probability that
a)none of the men selects his own hat?
b) exactly k of the men select their own hats?

It's difficult for me to think of this problem in the abstract...I always work better with actual numbers as opposed to variables.

Any help explaining this would be appreciated!
Thanks!
 
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  • #2
If you like numbers better, try it for N = 1, and then N = 2, and then N = 3, etc., until you see a pattern.

cookiemonster
 
  • #3
suppose there are four men, the probability none would select their own hat would
be
[tex] \frac34 * \frac23 *\frac12 [/tex]

this problem will be speed up with factorials where
4!= 4*3*2*1
3!= 3*2*1
etc
 
  • #4
Have you done the inclusion exclusion principle? If so then P(no hats correct) = 1-P(at least one hat correct) and the second of those is easy to work out if you know Inc-exc
 

1. What is the Matching Hats Problem?

The Matching Hats Problem is a mathematical puzzle that involves a group of n men and k hats. The goal is for each man to select a hat that matches one of the k hats selected by the group, without any man selecting the same hat as another.

2. What is the purpose of solving the Matching Hats Problem?

The purpose of solving the Matching Hats Problem is to exercise logical thinking and problem-solving skills. It also has real-world applications in fields such as cryptography and computer science.

3. What is the difficulty level of solving the Matching Hats Problem?

The difficulty level of solving the Matching Hats Problem varies depending on the number of men and hats involved. The more men and hats, the more complex the problem becomes.

4. Are there any known solutions to the Matching Hats Problem?

Yes, there are known solutions to the Matching Hats Problem. These solutions involve various mathematical algorithms and strategies, such as generating functions and combinatorics.

5. What are some tips for solving the Matching Hats Problem?

Some tips for solving the Matching Hats Problem include breaking the problem down into smaller parts, using trial and error to test different solutions, and thinking outside the box to find creative solutions. It can also be helpful to collaborate with others and discuss different approaches.

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