# bayes theorem and probability

A young engineer has invented holographic mobile phones and has approached a venture capital company to invest in it. The venture capital company considers the product to be an all or nothing product: either everyone will want one because everyone else has one or no one will want one because there will be no one to use it with. The company believes that the probability that it will take off netting them a profit of $2000000 is 0.14. If it doesn't take off then they expect that they would loose $200000. They are considering using a consumer survey to gather more information. However, the company has experience that shows that the probability that the consumer survey will predict success for a product that will fail is 0.24, and the probability that the consumer survey will predict failure when the product will be a success is 0.07. What is the monetary value of the information from a consumer survey to the venture capital company in this case? (ie what is the maximum that they should spend on a consumer survey)?

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#### Solution Preview

See attached for the decision tree

See the formulas in the cells for the decision tree.

Steps:

1. Use bayes theorem to calculate the posterior probabilities that the survey will forecast a success and failure.

2. Use the bayes theorem to calculate the probabilities that products ...

#### Solution Summary

A decision tree and its calculations are provided in this solution. The bayes theorem is discussed in respect to probability of investments.

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.