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.

• Specify the number (k) of mutually-exclusive events ( Ak ) that define the sample space.
• Enter values for P(Ak B) Or for P( Ak ) and P( B | Ak ).
• Click Calculate button to compute conditional probabilities P( Ak|B ).
 How many events (k) are in the sample space?

 EventAk Prob(Ak∩ B) Prob( Ak ) Prob( B | Ak ) Prob( Ak | B ) A1 Or A2

Summary Report

To create a report, enter data into the Bayes Rule 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.

• The sample space must consist of a set of k mutually-exclusive events - Ak.
• 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:

• The probability of the intersection of events Ak and B; that is, P(A B).
• 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 Dictionary available on this site. All of the terms used by the Bayes' Rule Calculator are defined in this online dictionary.

What if I don't understand the notation?

All of the notation used by the Bayes' Rule Calculator is defined below.

• k:
Number of mutually-exclusive events in the sample space
• P( Ak ):
Probability of event Ak
• P( B ):
Probability of event B
• P( B|Ak ):
Conditional probability of event B, given event Ak
• P( A1|B ):
Conditional probability of event A1, given event B
• P(Ak B):
Probability that event Ak and event B both occur, which is also known as the probability of the intersection of Ak and B.

Sample Problem

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 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.

• Event A1. It rains on Marie's wedding.
• Event A2. It does not rain on Marie's wedding
• Event B. The weatherman predicts rain.

In terms of probabilities, we know the following:

• P( A1 ) = 5/365 = 0.0136985 [It rains 5 days out of the year.]
• P( A2 ) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.]
• P( B | A1 ) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]
• 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) = P( A1 ) P( B | A1 ) P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 ) P(A1|B) = (0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.1) ] P(A1|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.