Question: In a study of a lake's fish population, scientists capture fish from the lake, then tag and release them. Suppose that over a period of five days, 200 fish of a certain type are tagged and released. As part of the same study, 20 such fish are captured three days later. Let X denote the number of tagged fish among the 20 captured. Suppose it is known that the lake has 1000 fish of this particular type.

(a) The distribution of the random variable X is (choose one)

(i) binomial

(ii) hypergeometric

(iii) negative binomial

(iv) Poisson.

(b) Use R commands to find P(X ≤ 4).

(c) Which distribution from those listed in part (a) can be used as an approximation to the distribution of X?

(d) Using the approximate distribution, give an approximation to the probability P(X ≤ 4), and compare it with the exact probability found in part (b).