The Hat Check Problem!

[Ahly]

Hi, for my IA I was thinking of doing the hat check problem. Essentially it goes like this; group of men walk into a restaurant checking in n number of hats at the door. The restaurant mixes the the hats up and as the evening is over, each man is randomly given a hat from the pile. What is the probability that no person gets their hat back?

Suprisingly, the answer is 1/e.

I am a math Sl student. I figured HL/further math students would likely recognize the problem. If anyone is familiar with the problem could you please elaborate on whether this is SL level or above. I have found some explanations rather complicated and wondered whether it was ultimately worth it given the amount of energy it would require with mocks coming up. I know I would be able to get my head around it, but have begun to doubt whether the return (in terms of marks) for this question would be worth it.

 I have time to change my topic so I am not particularly worried.

[kw0573]

As a former HL student, I have never seen this problem, but after some quick search on the web I see that it is not entirely out of HL syllabus but IB would not pose a question this way. 

Note that with n = 1, the probability is 1; 1/2 for n = 2; 1/3 for n = 3; 9/24 = 3/8 for n = 4. You can verify all this by listing possible outcomes. So 1/e is not always the probability. (Then what does 1/e represent?)

This topic is very manageable and very suitable for an HL student but for SL you do need to know slightly more about distributions, namely poisson distribution. I think if you have learned about SL Topic 5: Statistics and Probability then the IA should be manageable for a SL student in 10-15 hours if your teacher can help you with poisson distribution. It would be a very good idea to look for a topic of similar difficulty and familiarity if you choose not to pursue this idea.