Multiplication Rule Of Probability Calculator

Work out multiplication rule of probability instantly with clear inputs, formula shown and shareable results.

About Multiplication Rule Of Probability Calculator

The general multiplication rule is P(A ∩ B) = P(A)·P(B|A). Rearranging gives Bayes' theorem P(A|B) = P(A)P(B|A)/P(B), which reverses the direction of conditioning.

Formulas

Multiplication rule

P(A ∩ B) = P(A)·P(B|A)

Bayes' theorem

P(A|B) = P(A)·P(B|A) / P(B)

Worked example

With the default values below, Multiplication Rule Of Probability Calculator returns P(A and B) of 0.18000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Probability of Aaccepts 0.01–100 %: 60 %
Probability of B given Aaccepts 0–100 %: 30 %
Overall probability of Baccepts 0.01–100 %: 25 %

Results

P(A and B): 0.18000000
P(A given B) by Bayes: 0.72000000
Independence: Dependent — conditioning on A changes P(B)
P(A or B): 0.67000000

Step by step

1Values used
2Multiplication rule
3Bayes' theorem
4P(A and B)= 0.18000000
5P(A given B) by Bayes= 0.72000000
6Independence= Dependent — conditioning on A changes P(B)
7P(A or B)= 0.67000000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For P(A) = 0.6, P(B|A) = 0.3 and P(B) = 0.25?

Both occur with probability 0.18, and P(A|B) = 0.18/0.25 = 0.72.

When does it simplify?

Under independence P(B|A) = P(B), so the rule reduces to the plain product P(A)P(B).