Question

David's gasoline station offers 4 cents off per gallon if the customer pays in cash. Past evidence indicates that 40% of all customers pay in cash. During a one-hour period, 15 customers buy gasoline at this station. What is the probability that more than 8 and less than 12 customers pay in cash?

Answer 1

Answer:

The probability that more than 8 and less than 12 customers pay in cash is 0.0931.

Step-by-step explanation:

Let X = a customer at David's gasoline station pay in cash.

The probability of a customer paying in cash is, P (X) = p = 0.40

The number of customers at the gasoline station during a 1-hour period is,

n = 15.

Then the random variable X follows a binomial distribution, Bin (15, 0.40).

The probability function for a Binomial distribution is:

$P(X=x)={n\choose x}p^{x}(1-p)^{n-x}$

Compute the probability that more than 8 and less than 12 customers pay in cash as follows:

$P(8< X< 12)=P(X<12)-P(X<8)\\=P(X=9)+P(X=10)+P(X=11)\\=[{15\choose 9}(0.40)^{9}(1-0.40)^{15-9}]+[{15\choose 10}(0.40)^{10}(1-0.40)^{15-10}]\\+[{15\choose 11}(0.40)^{11}(1-0.40)^{15-11}]\\=0.0612+0.0245+0.0074\\=0.0931$

Thus, the probability that more than 8 and less than 12 customers pay in cash is 0.0931.