One strange phenomenon that sometimes occurs at U. S. airport security gates is that an otherwise law-abiding passenger is caught with a gun in his/her carry-on bag. Usually the passenger claims he/she forgot to remove the handgun from a rarely-used bag before packing it for airline travel. It’s estimated that every day 3,000,000 gun owners fly on domestic U. S. flights.

Suppose the probability a gun owner will mistakenly take a gun to the airport is 0.00001. What is the probability that tomorrow more than 35 domestic passengers will accidentally get caught with a gun at the airport?

Choose the closest answer.

a. 0.02
b. 0.91
c. 0.16
d. 0.84
e. 0.28

c. 0.16

Step-by-step explanation:

For each passenger there are only two possible outcomes. Either they are caught with a gun, or they are not. The probability of a passenger being caught with a gun is independent from other passengers. So we use the binomial probability distribution to solve this question.

However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that , .

In this problem, we have that:

So

What is the probability that tomorrow more than 35 domestic passengers will accidentally get caught with a gun at the airport?

This probability is 1 subtracted by the pvalue of Z when X = 35. So

has a pvalue of 0.8186

1 - 0.8186 = 0.1814.

The closest answer is:

c. 0.16