Problem 135E

Of the travelers arriving at a small airport, 60% fly on major airlines, 30% fly on privately owned planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those traveling on major airlines, 50% are traveling for business reasons, whereas 60%of those arriving on private planes and 90%of those arriving on other commercially owned planes are traveling for business reasons. Suppose that we randomly select one person arriving at this airport. What is the probability that the person

a is traveling on business?

b is traveling for business on a privately owned plane?

c arrived on a privately owned plane, given that the person is traveling for business reasons?

d is traveling on business, given that the person is flying on a commercially owned plane?

Solution 135E

Step1 of 5:

Let us consider the event “B” be the travelling on business.

Let event “M” be the fly major airline.

Let event “C” be the fly other commercial airline(not major).

Let event “R” privately owned plane.

Then the probabilities are given below:

P(M) = 0.60

P(R) = 0.30

P(C) = 0.10

P(B|M) = 0.50

P(B|R) = 0.60

P(B|C) = 0.90

Here our goal is:

a). We need to find the probability that person is traveling on business.

b). We need to find the probability that person is traveling for business on a privately owned plane.

c). We need to find the probability that person arrived on a privately owned plane, given that the person is traveling for business reasons.

d). We need to find the probability that person is traveling on business, given that the person is flying on a commercially owned plane.

Step2 of 5:

a).

Consider,

= 0.57

Hence, P(B) = 0.57.

Therefore, the probability that person is traveling on business is 0.57.

Step3 of 5:

b).

The probability that person is traveling for business on a privately owned plane is given by:

=

= 0.18

Hence, The probability that person is traveling for business on a privately owned plane is 0.18.

Step4 of 5:

c).

The probability that person arrived on a privately owned plane, given that the person is traveling for business reasons is given by:

=

[=0.18(Part(b)), =0.57(part(a))]

= 0.3157

Hence, The probability that person arrived on a privately owned plane, given that the person is traveling for business reasons is 0.3158.

Step5 of 5:

d).

The probability that person is traveling on business, given that the person is flying on a commercially owned plane is given by:

=

=

= 0.90

Hence, The probability that person is traveling on business, given that the person is flying on a commercially owned plane is 0.90.