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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.
Learning Objective

Apply the multiplication rule to the probability A and B occurring
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(AB).

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

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.

sample space
The set of all possible outcomes of a game, experiment or other situation.
Full Text
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(AB),
or alternatively as:
P(A∩B) = P(A) x P(BA).
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(AB) = P(A∩B) x P(B).
If P(AB) = 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
and
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(BA),
P(A∩B) =
Key Term Reference
 conditional probability
 Appears in this related concepts: Introduction to conditional probability (special topic), Defining conditional probability, and Conditional probability exercises
 experiment
 Appears in this related concepts: Experimental Design, Experiments, and Descriptive and Correlational Statistics
 independent
 Appears in this related concepts: Probability Histograms, Conditional Probability, and Party Identification
 independent event
 Appears in this related concepts: The Paradox of the Chevalier De Méré, Independence, and Mutually Exclusive Events
 probability
 Appears in this related concepts: Particle in a Box, The Addition Rule, and Theoretical Probability
 probability theory
 Appears in this related concepts: Applications of Statistics, Complementary Events, and Independence
 sample
 Appears in this related concepts: What Is a Confidence Interval?, Sampling, and Identify Product Benefits
Sources
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Cite This Source
Source: Boundless. “The Multiplication Rule.” Boundless Statistics. Boundless, 27 Jun. 2014. Retrieved 24 Apr. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/probability8/probabilityrules34/themultiplicationrule1714445/