# How do you think about probability questions?

I am curious about how other people think (model) probability questions?

Here is a sample question:

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 none of the men selects his own hat?

1. What is your first thought when you see this problem?
2. Do you see images in your head? If so, what kind of images show up in your head? Are there images of hats in a bag for example?
3. How do you model the problem?
4. Why do you choose your model?
5. What's your answer?
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I'm not convinced that this question is a good fit for math.SE, but I don't want to take unilateral action, so I'll wait and see what others think. This is more of a psychology question than a math question. –  Qiaochu Yuan May 18 '12 at 15:28
Is this a homework question? –  Graphth May 18 '12 at 15:48
I find this question troubling, however, there is a possibility of a misunderstanding. Would you care to clarify what is the origin and the aim of this question? Also, what you want to learn from the answers and how are you going to use them? This looks to me exactly like someone doing psychology experiment (or something similar) and judging from your previous questions this is not far fetched. –  dtldarek May 18 '12 at 17:52
I also vote to close this question as "off topic", but I will wait to see what others think since my vote is binding. –  Eric Naslund May 18 '12 at 19:38
it's not for a psychology experiment or anything of that sort. it's for my own education purposes. probability questions lend itself to being solved in many ways with different models. i think of generating sequences; my friend thinks "fractions"; some people try to relate to "toss a coin" or "flip a die" or "bag" models. i feel it's really important how you start a probability problem. i wanted to see the different approaches to solving the problem. –  Bonnie Yu May 19 '12 at 0:31

## 2 Answers

Imagine you have a refridgerator, a giraffe , a monkey, a dog, a cat, a mouse, a woman and an elephant together with enough food to feed them for a week. You also have access to a car, a ship and a tricycle.

Picture this: Put the giraffe on the elephant, the monkey on the giraffe, the dog on the monkey, the cat on the dog, the mouse on the cat.

The giraffe is afraid of the elephant, the elephant is afraid of the mouse, the mouse is afraid of the cat, the cat is afraid of the dog , the dog is afraid of the man and the man is afraid of the wife.

So if you are the giraffe, what do you do now ?

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I really appreciate the tongue-in-cheekiness of this answer. I hope that's what was intended. –  mixedmath May 23 '12 at 4:07

This is just a simple problem on derangement, which says that in how many ways can $n$ letters be put in $m$ envelopes, such that no letter goes into its correct envelope. And by simple inclusion-exclusion or using a recursion relation involving derangements of $n,n-1$ and $n-2$ objects, the answer comes out to be: $n!(\frac{1}{1!} - \frac{1}{2!} + \frac {1}{3!} - ... +- \frac {1}{n!})$. So, the required probability is actually $e^{-1}$ as $n \rightarrow \infty$.

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