Question

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

Answer 1

We assume that persons are independent. We consider $10$ independent random variables $X_i$, $i=1,\ldots,10$ and each of these variables has Bernoulli distribution. These variables correspond to randomly selected persons. Each variable takes value ('a person does not have a dog') with a probability $0.7$ and $1$ ('a person has a dog') with a probability $0.3$. Suppose that $a_1,a_2,\ldots,a_9$ are numbers such that $4$ of them are equal to $1$ and $5$ of them are . The probability is equal to: $p=\sum_{a_1,a_2,\ldots,a_9}P(X_1=a_1,X_2=a_2,\ldots,X_9=a_9,X_{10}=1)$, where the sum is taken over all numbers $a_1,a_2,\ldots,a_9$ such that $4$ of them are equal to $1$ and $5$ of them are . We point out that $P(X_1=a_1,X_2=a_2,\ldots,X_9=a_9,X_{10}=1)=(0.3)^5(0.7)^5$. The sum contains $C_{9}^4=\frac{9!}{5!4!}$ terms. Thus, $p=C_9^4(0.3)^5(0.7)^5\approx0.051$ (it is rounded to 3 decimal places)

Answer: $0.051$(it is rounded to 3 decimal places)