1. ## Hypergeometric Probability

Have a problem:

The sizes of animal populations are often estimated by using a capture-tag-recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y, the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k=4 animals are tagged and then released. A sample of n=3 animals is then selected at random from the same population. Find P(Y=1) as a function of N. What value of N will maximize P(Y=1).

Note that for the following [a,b] is the combinations rule such that this equals a!/b!(a-b)!.

I said to let X be the number of tagged animals among the n. X therefore is a hypergeometric probability distribution with r=4, y=1, n=3, and N=N. Therefore the distribution would be ([4,1]*[N-4,2])/[N,3].

Is this right and is this a function of N. How would you maximize this function?

2. Originally Posted by JaysFan31
Have a problem:

The sizes of animal populations are often estimated by using a capture-tag-recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y, the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k=4 animals are tagged and then released. A sample of n=3 animals is then selected at random from the same population. Find P(Y=1) as a function of N. What value of N will maximize P(Y=1).

Note that for the following [a,b] is the combinations rule such that this equals a!/b!(a-b)!.

I said to let X be the number of tagged animals among the n. X therefore is a hypergeometric probability distribution with r=4, y=1, n=3, and N=N. Therefore the distribution would be ([4,1]*[N-4,2])/[N,3].

Is this right and is this a function of N. How would you maximize this function?
This looks right to me. ([4,1]*[N-4,2])/[N,3] is P(Y=1), not the distribution. Writing out this probability as factorials, cancelling where possible, and ignoring any factor not having an N in it, I get P(Y=1) ~ (N-4)(N-5)/N(N-1)(N-2). This isn't nice to maximize using differentiation, so I plugged it into a spreadsheet and tried values. I found N = 10 was the maximizer. I did this in a hurry, so please confirm my calculations.

3. I got 12((N-4)(N-5)/N(N-1)(N-2)). I found the maximum of this to be 3. Can anyone confirm this?

4. Originally Posted by JaysFan31
I got 12((N-4)(N-5)/N(N-1)(N-2)). I found the maximum of this to be 3. Can anyone confirm this?
The value 3 cannot be either the maximizer N or the function value at the maximizer.

First, the formula would be invalid at N = 3 as it is not possible to take a sample without replacement of size 4 from a population of size 3.

Second, the formula is for a probability so its value is between 0 and 1.