A friend approaches you with the following problem:

"I have a drawer that contains N pairs of socks.  Each pair is a different
color, call them C1, C2, ... Cn.  These socks are all jumbled together.  I
cannot tell the color by feeling them.

Every morning, I close my eyes and draw two socks from the drawer.  
I put them back if they do not match. I keep doing this until I get
a matching pair, which I then wear.

Every day I count the number of tries it takes to get a matching pair.  If I
give you those counts for several days, can you tell me how many different
colors of socks I have?  That is, can you estimate N for me?  Can you give me
an idea of how likely it is that your estimate is correct?"

a) Write an expression for the probability of drawing a matching pair of socks
in one try.

b) Write an expression that gives the probability that the first match will
occur on the Kth try.

c) What is the expected value of this expression?

d) Using the method of moments, how will you estimate the number of diffeent
colored socks in the drawer?

e) How would you estimate the variance of that estimate, were you to repeat
the experiment multiple times.

f) Simulate this situation, and see how well your simulated data agrees with
your calculated expectations.

Write a program that allows you to sepcify the number of colors, and the number
of days.  For each day, draw socks (with replacement) until you get a match.
Estimate the number of colors and the standard error of the estimate.

Do this multiple times and find the mean and standard deviation of the
multiple estimates.

Draw conclusions.