Question

A lake contains 400 fish, and 100 of them are tagged by scientists. A researcher randomly catches 20 fish from the lake. (a) What is the probability that there are exactly 7 tagged fish? (b) What is the probability that the number of tagged fish is between 4 and 6? (c) Use binomial distribution to approximate the answer to part (a)

The answers are a) .1129 b) .5725 and c) .1124. Please explain how to solve the problem step by step.

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Answer #1

a) P(exactly 7 tagged fish) = 100C7 * 300C13/400C20 = 0.1129

b) P(tagged fish between 4 and 6) = 100C4 * 300C16/400C20 + 100C5 * 300C15/400C20 + 100C6 * 300C14/400C20 = 0.5725

c) n = 20

   p = 100/400 = 0.25

P(X = x) = nCx * px * (1 - p)n - x

P(X = 7) = 20C7 * (0.25)^7 * (0.75)^13 = 0.1124

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