Question

The probability that aperson living in a certaincity owns a dog is estimated to be 0.3. Find the probabilitythat the tenth person randomly interviewed inthat city is the fifth one to own a dog.

Answer 1

The probability that the tenth person randomly interviewed in that city is the fifth one to own a dog = 0.051

Given,

Consider the interview of a person is a trial.

Lets consider it success if a person  owns a dog and consequently , a person without a dog will be a failure.'

Let the random variable X represent the number of persons to be interviewed to get 5 dog owner: in other words, to get 5 successors.

Probability of a success in each trial is p = 0.3. Therefore probability of a failure in each trial is q = 1 - p = 0.7 .

Because trials are independent. X has a negative binomial distribution with parameters k = 5, p = 0.3

P(X =x ) = b(x: k, p) = $[(x-1)(k-1)] p^kq^r^-^k$

where, x = k , k + 1 , k + 2......(1)

Now, Lets find the probability that the tenth person randomly interviewed in that city is the fifth one to own a dog

Using equation (1), we get:

P(X = 10) = b(10: 5, 0.3)

                = [(10-1) (5 - 1)]$(0.3)^5(0.7)^1^0^-^5$

               = [(9)(4)] $(0.3)^5(0.7)^1^0^-^5$

               = 0.05146

Hence, The probability that the tenth person randomly interviewed in that city is the fifth one to own a dog = 0.051

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