Elk dwell in a certain forest. There are N elk, of which a simple random sample of size n are captured and tagged ("simple random sample" means that all (N C n) sets of n elk are equally likely). The captured elk are returned to the population, and then a new sample is drawn, this time with size m. This is an important method that is widely-used in ecology, known as capture-recapture. What is the probability that exactly k of the m elk in the new sample were previously tagged? (Assume that an elk that was captured before doesn't become more or less likely to be captured again.)

Solution: We can use the naive definition here since we’re assuming all samples of size m are equally likely. To have exactly k be tagged elk, we need to choose k of the n tagged elk, and then m − k from the N − n untagged elk. So the probability is
(n C k) x (N-n C m-k) / (N C m)
for k such that 0 is less than or equal to k, which is less than or equal to n, and 0 is less than or equal to m − k, which is less than or equal to N − n, and the probability is 0 for all other values of k (for example, if k > n the probability is 0 since then there aren't even k tagged elk in the entire population!). This is known as a Hypergeometric probability; we will encounter these probabilities again later in the course.

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