Your friend claims that he has ESP. In order to prove his claim to you, he is going to have your write the nam

Question

Your friend claims that he has ESP. In order to prove his claim to you, he is going to have your write the names of ten of your friends on slips of paper and put one name into each of 10 sealed envelopes. You will invite the 10 friends to your house and your friend with ESP will distribute the envelopes using his ESP powers.

Here is the question: How many people have to receive their own name before you believe that your friend has ESP and isn't just randomly giving out the envelopes?

To answer the question you will first need to simulate randomly giving envelopes with names to 10 people to see what the distribution of correct names under random distribution looks like. You must do as many simulations as you need to be sure about the distribution. You must also be clear about how you did the simulation and what the results were.

Next you will have to choose an alpha level for your test of ESP. How many of your friends must get their own names in the envelope before your ESP friend has performed that much better than random?

Answer 1

So what is the question?

Your teacher explicitly wants you to run the simulation to determine the distribution, rather than computing it directly (though it can be computed).

For the simulation you just need to generate a (long) sequence of random permutations of the numbers 1 - 10 and count the number of numbers that are left in their correct (i.e. original) places - the "fixed points" of the permutation.

Then you can say that out of N random permutations, K0 had no fixed points, K1 had 1 fixed point, K2 had 2 fixed points, etc.

Here are two sites with info on random permutations:
http://www.techuser.net/randpermgen.html
http://en.wikipedia.org/wiki/Random_permutation

I would use "Knuth's shuffle"

If you want to bypass the simulation (not a good idea), here is a page that can get you the data you need for the computation:
http://en.wikipedia.org/wiki/Random_permutation_statistics

Keep in mind that a "fixed point" (a number that the permutation does not change) is equivalent to a cycle of length 1.