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

the probability is 0.051 (5.1%)

Step-by-step explanation:

since the fact that person owns a dog independently of the behaviour of the other people, then the random variable X = number of people that owns a dog from 10 interviewed, follows a binomial distribution.

P (X=x) = C (n, x) * p^x * (1-p) ^ (n-x)

where

n = sample size = 10

p = probability that a person owns a dog = 0.3

x = number of people found that owns a dog = 5

C (n, x) = combinations of 5 persons who owns a dog from 10 interviewed (number of times we can observe 5 people from 10 owning a dog)

for our case we know that the tenth person has a dog, then considering that constraint the number of times observed must be modified and is equal to the number of times we can observe 4 people out of 9 that has a dog (since we know already that the tenth will be a person who owns a dog)

therefore

P (tenth person is the fifth one to own a dog) = C (n-1, x-1) * p^x * (1-p) ^ (n-x)

= C (9,4) * 0.3^5 * 0.7^ (10-5) = 0.051 (5.1%)

therefore the probability is 0.051 (5.1%)