100 trout are caught in a little lake and returned after they are tagged. Later, another 100 are caught and found to contain 7 tagged trout.
a) What is the probability of this is the lake conains n trout?
b) What is your best guess on n?
Let X be the random variable number of tagged trout in the second sample.
a) $\displaystyle \Pr(X = 7) = \frac{ {100 \choose 7} \, {n - 100 \choose 100 - 7}}{{n \choose 100}}$.
More generally, if m trout are caught and returned after they are tagged, and k trout are later caught and found to contain x tagged trout:
$\displaystyle \Pr(X = x) = \frac{ {m \choose x} \, {n - m \choose k - x}}{{n \choose k}}$.
b) The best estimate of n is the integer value of n that maximises Pr(X = 7).
Using technology is the easiest way of finding this value (although it's not difficult to do by hand if you use the right approach).