Bayes' Rule Calculator
The Bayes' Rule Calculator computes conditional probabilities P( Ak|B ),
based on known probabilities of other events. The calculator handles problems
that can be solved using
Bayes' rule.
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Specify the number (k) of mutually-exclusive events ( Ak
) that define the sample space.
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For each mutually-exclusive event, enter values for P(Ak
∩
B) Or for P( Ak ) and P( B | Ak ).
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Click the Calculate button to compute
conditional probabilities P( Ak|B ).
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To create a report, enter data into the Probability Calculator and click the Calculate button.
Instructions: To find the answer to a frequently-asked
question, simply click on the question.
What kinds of problems can the
Bayes' Rule Calculator handle?
The Bayes' Rule Calculator computes a
conditional probability, based on the values of related known
probabilities. Computations rely on Bayes' Rule.
The calculator can be used whenever Bayes' Rule can be applied.
Bayes' rule requires that the following conditions be met.
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The sample space must consist of a set of k
mutually-exclusive events - Ak.
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Within the sample space, there must exist an event B, for which the P(B) is not
equal to zero.
Bayes' rule also requires that you know certain probabilities.
For each event (Ak), you must know one of the following:
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The probability of the intersection of events Ak and B; that is, P(A
∩
B).
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The conditional probability of B given Ak and the probability of Ak;
that is, P( B|Ak ) and P( Ak ).
Note that for each event, you only need to know one of the above.
If you know P(A
∩
B), you don't need to know P( B|Ak ) and P( Ak
); and vice versa.
What are the meanings of the various
statistical terms used by the Bayes' Rule Calculator?
To use the Bayes' Rule Calculator and to understand the summary
report it prepares, you need to understand some statistical jargon. If you
encounter a term that you don't understand, visit the
Statistics Glossary. All of the terms used by the Bayes' Rule Calculator are
defined at the Help Center.
What if I don't understand the
notation?
Refer to the Notation sidebar
at the top of this web page. All of the notation used by the Bayes' Rule
Calculator is defined in the notation sidebar.
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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 mutually-exclusive events - it rains or it
does not rain. Additionally, a third event occurs when the weatherman predicts
rain. Notation for these events appears below.
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Event A1. It rains on Marie's wedding.
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Event A2. It does not rain on Marie's wedding
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Event B. The weatherman predicts rain.
In terms of probabilities, we know the following:
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P( A1
) = 5/365 = 0.0136985 [It rains 5 days out of the year.]
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P( A2
) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.]
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P( B | A1
) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]
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P( B | A2 ) = 0.1 [When it does not rain, the weatherman predicts
rain 10% of the time.]
We want to know P( A1 | 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( A1 | B ) =
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P( A1 ) P( B | A1 )
P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2
)
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| P( A1 | B ) =
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(0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ]
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| P( A1 | B ) =
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.111
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Note the somewhat unintuitive result. Even when the weatherman predicts rain, it
only 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.
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