**From:** Lee Corbin (*lcorbin@tsoft.com*)

**Date:** Sat Jul 05 2003 - 23:06:16 MDT

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Has anyone got more hard problems using Bayes' Theorem?

The ones in my textbooks aren't that challenging, e.g.,

#2 below. It was here on this list that the following

great problems were posted.

Any more?

Thanks,

Lee

1. It is known that about 1 woman in 100 gets breast cancer.

When a woman does have breast cancer, the test detects it

80 percent of the time. About one time in ten, however,

the test creates a false-positive; that is, someone who

does not have breast cancer nonetheless gets a positive

reading from the test. Given that a particular woman has

just scored positive on the test, what is the actual

probability that she has breast cancer?

2. If there is a burglary on a particular night the probability

is .99 that the burglar alarm will ring. If there is no

burglary on a particular night the probability is .005 that

the burglar alarm will ring, falsely. The probability is

.001 that a burglary will occur on a particular night.

What is the probability that there is a burglary, given

that the alarm just rang?

3. The probability that a newborn will have deformities

traceable to a sickness of its mother during pregnancy is 1%.

If a child is born healthy and normal, the probability that

the mother had rubella during her pregnancy is 10%. If a

child is born with deformities and it can be traced to some

sickness of the mother, the probability that the mother had

rubella during her pregnancy is 50%. What is the probability

that a child will be born with deformities if its mother had

rubella during her pregnancy?

4. (and Eliezer's egg problem, which I don't have ready to hand).

5. A nurse has just started to count the babies in a hospital

nursery. She has just counted that there are two boys, and has

not counted the girls, when, at 11:00, a new baby is brought

in to the nursery. A baby is then selected at random, from

all the babies present, to have its footprint taken. The

selected baby happens to be a boy. What is the probability

that the baby added at 11:00 was a girl?

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