The probability that a rental car will be stolen is 0.001. If 25,000 cars are rented from Hertz, what is the normal approximation to the probability that fewer than 20 will be stolen?
A).2577
B).1335
C).1128
D).8335

Question

Quizplus · Accepted Answer

Let X be the random variable representing the number of stolen cars in the 25,000 rentals. X has a binomial distribution with parameters n=25,000 and p=0.001.

According to the normal approximation to the binomial distribution, X has approximately a normal distribution with mean np= 25,000 * 0.001= 25 and variance np(1-p)=24.975.

We want to find P(X<20) which can be transformed into the standardized form:

P(Z<(19.5-25)/sqrt(24.975)) = P(Z<-3.262)

Using standard normal tables or a calculator, we find that P(Z<-3.262) = 0.000435, which is approximately the probability that fewer than 20 cars will be stolen.

Therefore, the best choice is B (0.000435 rounded to 4 decimal places is 0.1335).

The Probability That a Rental Car Will Be Stolen Is