Question

At a gas station, suppose that 45% of the customers purchase premium grade gas. Assume that these customers decide independently. Find the probability that at least one of the next three customers purchases premium gas. (Hint: Think about the complement.)

Answer 1

0.29

Explanation:

Let the three cases be A, B ,C

P(A U B U C) = P(A) + P(B) + P(C)

Probability that one of the customers purchases premium gas, P(A) = 0.45(1-0.45)(1-0.45)=0.09

Probability that two of the customers purchases premium gas, P(B) ={0.45) (0.45)(1-0.45)=0.11

Probability that the customers purchases premium gas, P(C)= (0.45)(0.45)(0.45)=0.09

P(A) + P(B) + P(C) = 0.09+0.09+0.11=0.29

Answer 2

The answer to the question is

The probability that at least one of the next three customers purchases premium gas is the complement of the probability that none of the next three customers purchase premium gas = 1 - (1-P(A))³ = 0.834

Explanation:

The probability that a customer would purchase premium grade = 45 %

That is P(A) = 0.45 and

The probability that the customer would purchase another grade = P(B) = 0.55

Therefore the probability of at least one of the next three customers purchase premium gas is

P(k=0) = (1 - P)ⁿ and the probability of at least one customer purchases premium gas is the compliment of the probability that the next three customers purchase another gas brand

that is (1 - P(A))×(1 - P(A))×(1 - P(A)) = P(B)×P(B)×P(B) = 0.55³ and the complement is 1 - 0.55³ = 0.834