Bayes worksheet

Due 5/3/99

Note: p = probability

Bayes theorem:

p(target event, given cue) =

p(cue, given target event)*p(target event)

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p(cue, given target)*p(target event) + p(cue, given no target event)*p(no target event)

1 in 10 high school students get into prestigious colleges. 1 in 5 high school students take prep courses for college entrance tests. Of those in prestigious colleges, 1 in 20 took prep courses. Of those in non-prestigious colleges, 22 out of 100 took prep courses.

Suppose you want to answer the following question:

What is the chance of getting into a prestigious college, given you took a prep course?

1. What is the target event?


getting into a prestigious college

2. What is the cue?


taking a prep course

3. What is the probability of the cue, given the target event?


.05

4. What is the probability of the target event overall?


.10

5. What is the probability of the cue, given the target event doesn=t occur?


.22

6. What is the probability of the target event not occurring overall?


.9

7. Use your answers 3-6 above to compute the answer to the question:

What is the chance of getting into a prestigious college, given you took a prep course?

(.05)*(.10)/(.05)*(.10)+(.22)*(.90) =
.005/(.005 + .198) =
.02

8. If you know that someone took a college prep course, does it provide any diagnostic information about their likelihood of getting into a prestigious college? If so, what does it predict?


The probability of getting into a prestigious college among those who took prep classes is LESS (2 in 100) than the probability of getting into a prestigious college among the general public (10 in 100). [Keep in mind we are not making asssumptions about causality -- taking the course doesn't necessarily render people dumber or less desirable to admissions committees. The group of students who take these courses may be different from the kinds of students who don't.]