Bayes’ Rule:
1
()
*(
)
)
jj
j
N
ii
i
PA
PB
PA PB
=
 Α
Β =
 Α
∑
It might look ugly, but it is very useful. The only thing is you have to correctly setup the
events. For example, the Ai’s are a partition of your sample space. Most Bayes problems
will have 2 partitions of the sample space. Therefore you have to choose the one partition
that you are using based on the above formula.
Example from Class:
We have “Disease D” that affects 15% of the population. Suppose the test for Disease D
turns out + 95% of the time you actually have Disease D. Suppose further that the test for
Disease D turns out – 90% of the time you do not have Disease D.
There are 2 natural partitions in this problem. One is do you or do you not have
Disease D. The other is did you test positive or negative. Let D be the event that you have
Disease D. Let O be the event that you test positive. In relation to the Bayes’ formula
above, would D and
be the partition (i.e. would D and
act as the Ai’s) or would
O and
be the partition?
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 Spring '08
 MARTIN
 Conditional Probability, Probability theory, 15%, 60%, 90%

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