Bus 437: Decision Analysis

BUS 437: Decision Analysis

Practice # 1 - Solution

1. Suppose that a doctor performs a test to determine if one of her patients is pregnant. The test gives either a positive or negative result. Most of the time the test is quite accurate: only 1 positive result in 100 is incorrect and only 1 negative result in 100 is incorrect. Given that the doctor has found that about 70% of women who come to her for pregnancy tests are in fact pregnant, use Bayes' Rule to calculate the probability a woman the doctor sees is pregnant given that the pregnancy test gives a positive result.

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1. Boy or Girl? Suppose that the probability that a child, born to a couple, will be male or female varies depending on the genetic characteristics of the couple.  Assume further that there are three types of couples -- labeled, A, B, and C.  For type A couples, the probability that any child they have will be male is 0.8.  For types B and C the corresponding probabilities are 0.5 and 0.2. Assume for any couple that successive births are independent of each other with respect to the sex of the child (in other words, for a type A couple, the probability of a boy remains at 0.8 regardless of the sex of previous children that they might of had). Because it is virtually impossible to determine in advance which type a particular couple is, all that physicians can rely on when predicting the chances of a boy or girl in any pregnancy are population statistics.  15% of all couples are type A, 70% type B and 15% type C.

1. What is probability that a randomly selected couple will have a boy as their first child?
2. If a particular couple’s first born child is a boy, which type of couple are they most likely to be?  What is the probability that their next child (should they choose to have one) will be a boy?
3. Suppose the couple in part (a) does have a second boy, what then is the likelihood that they are a type A couple?

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1. Joe Hinkle, the owner of a small manufacturing plant, faces an important decision. Joe has just been granted a patent for GazMizer, an electronic device that can be installed in automobiles to improve their efficiency and performance. Pilgrim Automotive Supplies has offered Joe $400,000 for the patent. Alternatively, Clayton Luxury Automobiles has proposed a joint agreement under which Clayton would offer the GazMizer as an option on its new cars, and Joe would manufacture the part exclusively for Clayton. Joe is uncertain about how many of Clayton's customers will order the GazMizer for their cars. Joe expects to net $1.3 million in profit if their response is favorable, but to lose $50 thousand if their response is unfavorable. Joe believes that the (prior) probability of a favorable response is 60%. Suppose that, in making business decisions, Joe always seeks to choose that strategy that maximizes his expected monetary value (EMV).

While considering these options, Joe sees an advertisement for Flo, a market researcher who specializes in the luxury car market. Flo offers to conduct, for a fee of $50 thousand, a survey to predict whether the market for GazMizer will be favorable. Joe has data on Flo's past success in predicting the market for new products. Considering just those products for which her prediction was recorded, she correctly had predicted market success ("Good") for 90% of the products that turned out to be successful (Good). On the other hand, she also had predicted "Good" for 25% of the products that turned out to be market failures (Poor).