Bayes Theorem (aka, Bayes Rule)
Bayes' theorem (also known as Bayes' rule) is a useful tool for calculating
conditional probabilities. Bayes' theorem can be stated as follows:
Bayes' theorem. Let A
_{1}, A
_{2},
... , A
_{n} be a set of mutually exclusive events that together form
the sample space S. Let B be any event from the same sample space, such that
P(B) > 0. Then,
P( A_{k}  B ) =

P( A_{k}
∩ B )
P( A_{1} ∩
B ) + P( A_{2} ∩
B ) + . . . + P( A_{n} ∩ B )

Note: Invoking the fact that P( A
_{k} ∩
B ) = P( A
_{k} )P( B  A
_{k} ), Baye's theorem can
also be expressed as
P( A_{k}  B ) =

P( A_{k} ) P( B  A_{k} )
P( A_{1} ) P( B  A_{1} ) + P( A_{2} ) P( B  A_{2}
) + . . . + P( A_{n} ) P( B  A_{n} )

Unless you are a worldclass statiscian, Bayes' theorem (as expressed above) can
be intimidating. However, it really is easy to use. The remainder of this
lesson covers material that can help you understand when and how to apply
Bayes' theorem effectively.
When to Apply Bayes' Theorem
Part of the challenge in applying Bayes' theorem involves recognizing the types
of problems that warrant its use. You should consider Bayes' theorem when the
following conditions exist.

You know at least one of the two sets of probabilities described below.

P( A_{k} ∩
B ) for each A_{k}

P( A_{k} ) and P( B  A_{k} ) for each A_{k}
Bayes Rule Calculator
Use the Bayes Rule Calculator to compute conditional probability, when
Bayes' theorem can be applied. The calculator is free, and it is easy to use.
It can be found under the Stat Tools
tab, which appears in the header of every Stat Trek web page.
Sample Problem
Bayes' theorem can be best understood through an example. This section presents
an example that demonstrates how Bayes' theorem can be applied effectively to
solve statistical problems.
Example 1
Marie is getting married tomorrow, at an outdoor ceremony in the desert. In
recent years, it has rained only 5 days each year. Unfortunately, the
weatherman has predicted rain for tomorrow. When it actually rains, the
weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he
incorrectly forecasts rain 10% of the time. What is the probability that it
will rain on the day of Marie's wedding?
Solution: The sample space is defined by two mutuallyexclusive events 
it rains or it does not rain. Additionally, a third event occurs when the
weatherman predicts rain. Notation for these events appears below.

Event B. The weatherman predicts rain.
In terms of probabilities, we know the following:

P( B  A_{2} ) = 0.1 [When it does not rain, the weatherman predicts
rain 10% of the time.]
We want to know P( A_{1}  B ), the probability it will rain on the day
of Marie's wedding, given a forecast for rain by the weatherman. The answer can
be determined from Bayes' theorem, as shown below.
P( A_{1}  B ) =

P( A_{1} ) P( B  A_{1} )
P( A_{1} ) P( B  A_{1} ) + P( A_{2} ) P( B  A_{2}
)

P( A_{1}  B ) =

(0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ]

P( A_{1}  B ) =

0.111

Note the somewhat unintuitive result. Even when the weatherman predicts rain, it
rains only about 11% of the time. Despite the weatherman's gloomy
prediction, there is a good chance that Marie will not get rained on at her
wedding.