# More Hard Problems Using Bayes' Theorem, Please

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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