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Multiplication Rule of Probability
The multiplication rule of probability defines the condition between two given events. For two events, A and B associated with a sample space S, A∩B denotes the events in which both events have occurred. This is also known as the multiplication theorem in probability. The probabilities of the two given events are multiplied to give the probability of those events occurring simultaneously.
What is Multiplication Rule of Probability?
Multiplication rule of probability states that whenever an event is the intersection of two other events, that is, events A and B need to occur simultaneously. Then, P(A and B)=P(A)⋅P(B). The set A∩B denotes the simultaneous occurrence of events A and B, that is the set in which both events A and event B have occurred. Event A∩B can be written as AB. The probability of event AB is obtained by using the properties of conditional probability, which is given as P(A ∩ B) = P(A) P(B  A).
Multiplication Rule of Probability for Dependent Events
If the outcome of one event affects the outcome of the other, then those events are referred to as dependent events. Sometimes, the occurring of the first event impacts the probability of the second event. From the theorem, we have, P(A ∩ B) = P(A) P(B  A), where A and B are independent events.
Multiplication Rule of Probability for Independent Events
If the outcome of one event does not affect the outcome of another, then those events are referred to as independent events. The multiplication rule of probability for dependent events can be extended for the independent events. We have, P(A ∩ B) = P(A) P(B  A), so if the events A and B are independent, then, P(B  A) = P(B), and thus, the above theorem reduces to P(A ∩ B) = P(A) P(B). That means that the probability that both of these occur simultaneously is the product of their respective probabilities.
Multiplication Rule of Probability Formula
The multiplication rule of probability states that the probability of the events, A and B, both occurring together is equal to the probability that B occurs times the conditional probability that A occurs given that B occurs.
 The multiplication rule can be written as P(A∩B)=P(B)⋅P(AB).
 The general multiplication rule of probability can be obtained in a simple way, just multiplying both sides of the conditional probability equation by the denominator.
Multiplication Rule of Probability Proof
The probability of the intersection of two events, A and B is obtained by using the properties of conditional probability.
 We know that the conditional probability of event A given that B has occurred is denoted by P(AB) and is given by: P(AB) = P(A∩B)P(B), where, P(B)≠0. P(A∩B) = P(B)×P(AB) …….(1)
 P(BA) = P(B∩A)P(A),where, P(A) ≠ 0. P(B∩A) = P(A)×P(BA)
 Since, P(A∩B) = P(B∩A), P(A∩B) = P(A)×P(BA) ……..(2)
 From (1) and (2), P(A∩B) = P(B)×P(AB) = P(A)×P(BA), P(A) ≠ 0,P(B) ≠ 0. Hence, the result so obtained is known as the multiplication rule of probability.
 For independent events A and B, P(BA) = P(B). The equation (2) can be modified as, P(A∩B) = P(B) × P(A)
Multiplication Rule of Probability for n Events
Now, to obtain the multiplication rule of probability for n Events, the extension of the multiplication theorem of probability to n events for n events A_{1}, A_{2}, … , A_{n}, we have P(A_{1} ∩ A_{2} ∩ … ∩ A_{n}) = P(A_{1}) P(A_{2}  A_{1}) P(A_{3}  A_{1} ∩ A_{2}) … × P(A_{n} A_{1} ∩ A_{2} ∩ … ∩ A_{n1})
For n independent events, the multiplication theorem reduces to P(A_{1} ∩ A_{2} ∩ … ∩ A_{n}) = P(A_{1}) P(A_{2}) … P(A_{n}).
Related Topics
The following related topics help in better understanding the multiplication rule of probability.
Multiplication Rule of Probability Examples

Example 1: What is the probability of getting a 5 and then a 2 with the normal 6sided die?
Solution:
Sample space = {1, 2, 3, 4, 5, 6}
Total events = 6
Probability of getting a 5 = 1/6
Probability of getting a 6 = 1/6
Applying the multiplication rule of probability for independent events,
P(getting a 5 and then a 2 ) = (1/6).(1/6) = 1/36.
Therefore, the probability of getting a 5 and then a 2 with the normal 6sided die is 1/36.

Example 2: Two cards are selected without replacing the first card from the deck. Find the probability of selecting a king and then selecting a queen.
Solution:
Total events = 52
Since the first card is not replaced, the events are dependent.
Probability of selecting a king = P(K) = 4/52
Probability of getting a queen = P(Q) = 4/51 (one card drawn first has not been replaced)
P(a king & a then queen) = P(K).P(QK)
=4/52 . 4/51 = 16/2652 = 1/166.
Therefore, the probability of selecting a king and then selecting a queen is 1/166.
FAQs on Multiplication Rule of Probability
What Is the Multiplication Theorem of Probability?
As per the multiplication theorem of probability, the probability of simultaneous occurrence of two events A and B is the product of the probability of the other, given that the first one has occurred. This is called the Multiplication Theorem of probability.
Why Do We Use the Multiplication Rule in Probability?
Using the multiplication rule, we can calculate the probability that events A and B occurring jointly given that events, A and event B occur individually.
How Do you Find the Probability of 3 Events Using Multiplication Rule of Probability?
In the case of three events, A, B, and C, the multiplication rule is given as, the probability of the intersection P(A and B and C) = P(A)P(BA)P(CA and B).
How To Use the Multiplication Rule of Probability?
In case, there are two events occurring simultaneously, just multiply the probability of the first event by the second. For example, if the probability of event A is 2/7 and the probability of event B is 5/7 then the probability of both events happening at the same time is calculated using the multiplication rule of probability, i.e, (2/7)*(5/7) = 10/49.
The Multiplication Rule Is Used To Calculate What Type of Probability?
The multiplication rule calculates the probability of multiple events happening together using the known probabilities of the individual events.
What Is Multiplication Rule of Probability for Dependent Events?
In the case of dependent events, by applying the multiplication rule of probability, the probability of events is given by P(A and B)=P(A)⋅P(B  A), where A and B occur simultaneously
What Is Multiplication Rule of Probability for Independent Events?
When A and B are two independent events, then as per the multiplication rule, the probability that both events occur simultaneously is given as P(A and B)=P(A)⋅P(B)
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