# Probability and Bayes' Theorem

33. In a survey of MBA students, the following data were obtained on students' first reason for application to the school in which they matriculated.

Reason for Application

School Quality School cost or convenience Other Total

Full Time

Part Time 421

400 393

593 76

46 890

1039

Total 821 986 122 1929

a. Develop a joint probability table for these data

b. Use the marginal probabilities of school quality, school cost or convenience, and other to comment on the most important reason for choosing a school.

c. If a student goes full time, what is the probability that school quality is the first reason for choosing a school?

d. If a student goes part time, what is the probability that school quality is the first reason for choosing a school?

e. Let A denote the event that a student is full time and let B denote the event that the student lists school quality as the first reason for applying. Are events A and B independent? Justify your answer.

39. The prior probabilities for events A1 and A2 are P(A1)=.40 and P(A2)=.60. It is also knows that P(A1 . Suppose P(B|A1)=.20 and P(B|A2)=.05

a. Are A1 and A2 mutually exclusive? Explain.

b. Compute P(A1 and P(A2

c. Compute P(B)

d. Use Bayes' theorem to compute P(A1|B) and P(A2|B)

43. Small cars get better gas mileage, but they are not as safe as bigger as cars. Small cars accounted for 18% of the vehicles on the road, but accidents in involving small cars led to 11,898 fatalities during a recent year (Reader's Digest, May 2000). Assume the probability a small car is involved in an accident is .18. The probability of an accident involving a small car leading to a fatality is .128 and the probability of an accident involving a fatality. What is the probability a small car was involved? Assume that the likelihood of getting into an accident is independent of car size.

45. In an article about investment growth, Money magazine reported that drug stocks show powerful long-term trends and offer investors unparalleled potential for strong and steady gains. The federal Health Care Financing Administration supports this conclusion through its forecast that annual prescription drug expenditures will reach $366 billion by 2010, up from $117 billion in 2000. Many individuals age 65 and older rely heavily on prescription drugs. For this group, 82 % take prescription drugs regularly, 55% take three or more prescription regularly, and 40% currently use five or more prescriptions In contrast, 49% of people under age 65 take prescriptions regularly, with 37% taking three or more prescriptions regularly, with 37% taking three or more prescriptions regularly and 28% using five or more prescriptions (Money, September 2001). The U.S Census Bureau reports that of the 281,421,906 people in the United States, 34,991,753 are age 65 years and older (U.S Census Bureau, Census 2000).

a. Compute the probability that a person in the United States is age 65 or older.

b. Compute the probability that a person takes prescription drugs regularly

c. Compute the probability that a person is age 65 or older and takes five or more prescriptions.

d. Given a person uses five or more prescriptions, compute the probability that a person is age 65 or older.

https://brainmass.com/statistics/bayesian-probability/probability-and-bayes-theorem-174516

#### Solution Preview

33. In a survey of MBA students, the following data were obtained on students' first reason for application to the school in which they matriculated.

Reason for Application

School Quality School cost or convenience Other Total

Full Time 421 393 76 890

Part Time 400 593 46 1039

Total 821 986 122 1929

a. Develop a joint probability table for these data

We develop a joint probability table by dividing all the numbers by the total number= 1929

School Quality School cost or convenience Other Total

Full Time 0.22 0.20 0.04 0.46

=421/1929 =393/1929 =76/1929 =0.22+0.2+0.04

Part Time 0.21 0.31 0.02 0.54

=821/1929 =986/1929 =122/1929 =0.21+0.31+0.02

Total 0.43 0.51 0.06 1.00

=0.22+0.21 =0.2+0.31 =0.04+0.02

b. Use the marginal probabilities of school quality, school cost or convenience, and other to comment on the most important reason for choosing a school.

Marginal probability:

School Quality= 0.43

School cost or convenience= 0.51

Other= 0.06

The most important reason is School Cost or convenience

c. If a student goes full time, what is the probability that school quality is the first reason for choosing a school?

Probability for full time= 0.46

Probability for full time and School Quality= 0.22

Therefore, if a student goes full time, the probability that school quality is the first reason for choosing a school= 0.48 =0.22 / 0.46

d. If a student goes part time, what is the probability that school quality is the first reason for choosing a school?

Probability for part time= 0.54

Probability for part time and School Quality= 0.21

Therefore, if a student goes part ...

#### Solution Summary

Solves probability questions involving conditional probability, Bayes' Theorem.