Problem 41 - A First Course in Probability

Question

A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the \(20 ?\) What assumptions are you making?

Step-by-Step Solution

Answer

In conclusion, when catching 20 fish from a pond with 100 fish out of which 30 are carp, we can model this with a hypergeometric distribution having a mean number of carps of 6 and a variance of approximately 2.8282, under the assumption that each fish has an equal chance of being caught independently.

Step 1: Identify the hypergeometric distribution

The catching of fishes can be modeled as a hypergeometric distribution, where we have a population of size \(N=100\), of which \(K=30\) are carp. A sample of size \(n=20\) is chosen, and we want to find the mean and variance of the number of carp among these 20 fish caught.

Step 2: Calculate the mean

Mean of a hypergeometric distribution is given by: Mean (\(E[X]\)) = \(n \times \frac{K}{N}\) Where, \(n\) = number of fish caught \(K\) = number of carp in the pond \(N\) = total number of fish in the pond Mean (\(E[X]\)) = \(20 \times \frac{30}{100}\) = \(6\) So, the mean number of carps in the 20 fish caught is 6.

Step 3: Calculate the variance

Variance of a hypergeometric distribution is given by: Variance (\(Var[X]\)) = \(n \times \frac{K}{N} \times \frac{(N-K)}{N} \times \frac{(N-n)}{(N-1)}\) Where, \(n\) = number of fish caught \(K\) = number of carp in the pond \(N\) = total number of fish in the pond Variance (\(Var[X]\)) = \(20 \times \frac{30}{100} \times \frac{(100-30)}{100} \times \frac{(100-20)}{(100-1)}\) = \(20 \times \frac{30}{100} \times \frac{70}{100} \times \frac{80}{99}\) Variance (\(Var[X]\)) = \(2.8282\) So, the variance of the number of carps in the 20 fish caught is approximately 2.8282. In conclusion, when catching 20 fish from the pond, the mean number of carps among them is 6, and the variance is approximately 2.8282, under the assumption that each fish has an equal chance of being caught and we catch each fish independently.