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Multiplication Theorem of Probability

 

Whenever we need to define a particular condition between two events, we use the multiplication theorem of probability. The multiplication theorem of probability is related to the set of events that have occurred simultaneously. Thus, using the multiplication theorem, we can find the probability of two events occurring simultaneously.

Definition

Whenever we find an intersection between two events, then we use the multiplication theorem. Suppose we have two events, A and B. Then the intersection of these two events is denoted by A ∩ B, only if these two events occur simultaneously. Therefore,

P (A and B) = P (A) ⋅ P (B) = P (A ∩ B)

We can denote the occurrence of these events using conditional probability as well. For instance, A has occurred before B, then the probability of both events occurring simultaneously is given by:

P (A ∩ B) = P (A) P (B | A)

Multiplication rule for dependent events

If the probability of one event changes when the other event’s probability changes, then those events are known as dependent events. For example, students have to play football depending on whether the rain will happen or not. Therefore, the probability of playing the football game is dependent on the occurrence of rain. Hence, both these events are dependent.

Also, we can find the probability of playing football even if rain has occurred. This means one event is completed. Therefore, using conditional probability,

P (A ∩ B) = P (A) P (B | A)

Multiplication rule for independent events

If one event remains unchanged or unaffected even though the other event is occurring, these events are known as independent events. Thus, we can find the probability of occurrence of events even if they are independent of each other.

Consider an event of the happening of a cricket match, and consider another event of a football match. Since these two events are different, therefore, they are independent events. We can find the probability of these events using the multiplication rule.

According to the multiplication rule, the probability of two dependent events is given by:

P (A ∩ B) = P (A) P (B | A)

Since A and B are independent, which means B and A will have no relation. In addition, there will be no application of conditional probability in this case. Therefore, the probability of two independent events is given by:

P (A ∩ B) = P (A) . P (B)

Proof of multiplication rule

We know that the conditional probability of an event A when B has occurred according to the multiplication theorem of probability is denoted by:

P (A | B) = P (A∩B) . P (B)

where P (B) ≠ 0

P (A∩B) = P (B) × P (A | B)                                                       (1)

P (B | A) = P (B ∩ A) . P (A)

where, P(A) ≠ 0

P (B ∩ A) = P (A) × P (B | A)

Since, P (A ∩ B) = P (B ∩ A), we have, P (A ∩ B) = P (A) × P (B | A)  (2)

From above equations (1) and (2), we have,

P (A ∩ B) = P (B) × P (A | B) = P (A) × P (B | A)

where, P (A) ≠ 0, given that, P (B) ≠ 0

Hence, the above result is the multiplication theorem of probability.

For independent events A and B, P(B|A) = P(B). Therefore, we can modify equation (2) as P(A∩B) = P(B) × P(A) for independent events.

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