# The Multiplication Rule

## The multiplication rule states that the probability that A and B occur is equal to the probability that A occurs times the probability that B occurs.

#### Key Points

• The multiplication rule is understood given that we know A has already occurred.

• The multiplication rule can be written as: P(A∩B) = P(B) x P(A|B).

• The events A and B are defined on a sample space, which is the set of all possible outcomes of an experiment.

• We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator.

#### Terms

• The probability that A and B occur is equal to the probability that A occurs times the probability that B occurs, given that we know A has already occurred.

• The set of all possible outcomes of a game, experiment or other situation.

In probability theory, The Multiplication Rule states that the probability that A and B occur is equal to the probability that A occurs times the probability that B occurs, given that we know A has already occurred. This rule can be written:

P(A∩B) = P(B) x P(A|B),

or alternatively as:

P(A∩B) = P(A) x P(B|A).

As illustrated in , the events A and B are defined on a sample space, which is the set of all possible outcomes of an experiment.

We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator. In general, the conditional probability can be defined as follows:

If P(B) ≠ 0 then the conditional probability of A relative to B is given by

P(A|B) = P(A∩B) x P(B).

If P(A|B) = P(A) then we say that A and B are independent events.

As an example, suppose that we draw two cards out of a deck of cards and let A = {first card is an ace}, and B = {second card is an ace}, then

$P(A)=\frac { 4 }{ 52 }$,

and

$P\left( { B }|{ A } \right) =\frac { 3 }{ 51 }$.

The denominator in the latter equation is 51 since we know a card has been drawn already. Therefore, there are 51 left in total. We also know the first card was an ace, therefore:

P(A∩B) = P(A) x P(B|A),

P(A∩B) = $\frac { 4 }{ 52 } \times \frac { 3 }{ 51 } =0.0045$.

#### Key Term Glossary

conditional probability
the probability that an event will take place given the restrictive assumption that another event has taken place, or that a combination of other events has taken place
##### Appears in these related concepts:
experiment
A test under controlled conditions made to either demonstrate a known truth, examine the validity of a hypothesis, or determine the efficacy of something previously untried.
##### Appears in these related concepts:
independent
not dependent; not contingent or depending on something else; free
##### Appears in these related concepts:
independent event
the fact that A occurs does not affect the probability that B occurs
multiplication rule
The probability that A and B occur is equal to the probability that A occurs times the probability that B occurs, given that we know A has already occurred.
##### Appears in these related concepts:
probability
The relative likelihood of an event happening.
##### Appears in these related concepts:
probability theory
The mathematical study of probability (the likelihood of occurrence of random events in order to predict the behavior of defined systems).
##### Appears in these related concepts:
sample
a subset of a population selected for measurement, observation, or questioning to provide statistical information about the population
##### Appears in these related concepts:
sample space
The set of all possible outcomes of a game, experiment or other situation.