A zoo wishes to catch some animals of a certain species. For each animal caught,

Question

A zoo wishes to catch some animals of a certain species. For each animal caught, the probability that it is a male is p and a female is q = 1 - p. Each animal roams around the search area independently.
a. Find the probability that the first male/female pair is caught exactly after the zoo captures the fifth animal.
b. How many animals are expected to be caught before the first male/female pair is found? What is the value of p such that this expected value is a minimum?
c. How many animals are expected to be caught before the second male/female pair is found? What is this value when p = q = 0.5?

Explanation / Answer

A zoo wishes to catch some animals of a certain species. For each animal caught, the probability that it is a male is p and a female is q = 1 - p. Each animal roams around the search area independently.

a. Find the probability that the first male/female pair is caught exactly after the zoo captures the fifth animal.

So the first four animals are the same gender (w/o loss of generality, assume the first four animals are all male), and that the fifth is a male.

Therefore, the probability is (0.5)^4 = 0.0625

b. How many animals are expected to be caught before the first male/female pair is found? What is the value of p such that this expected value is a minimum?

In both cases, the probability is p = 0.5 = q = 0.5. In this way, it allows for the quickest selection of one of each (because the probability of each is exactly one-half).

c. How many animals are expected to be caught before the second male/female pair is found?

In the example from (a), 5 more.

What is this value when p = q = 0.5?

If p = q = 0.5, 4 animals are to be caught before we have two pairs

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