Question

Forty-four percent of customers who visit a department store make a purchase. What is the probability that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase?

A. .1070

B. .1752

C. .8930

D. .0033

Answer 1

Answer: A. .1070

Explanation:

We would apply the formula for binomial distribution which is expressed as

P(x = r) = nCr × p^r × q^(n - r)

Where

x represent the number of successes.

p represents the probability of success.

q = (1 - r) represents the probability of failure.

n represents the number of customers sampled.

From the information given,

p = 44% = 44/100 = 0.44

q = 1 - p = 1 - 0.44

q = 0.56

n = 9

x = r = 6

Therefore,

P(x = 6) = 9C6 × 0.44^6 × 0.56^(9 - 6)

P(x = 6) = 84 × 0.0073 × 0.175616

P(x = 6) = 0.107

Answer 2

A. .1070

Explanation:

For each customer, there are only two possible outcomes. Either they make a purchase, or they do not. The probability of a customer making a purchase is independent from other customers. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

Forty-four percent of customers who visit a department store make a purchase.

This means that
`p = 0.44`

What is the probability that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase?

This is
`P(X = 6)` when n = 9. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 6) = C_(9,6).(0.44)^(6).(0.56)^(3) = 0.1070`

So the correct answer is:

A. .1070