Multiplication Theorem of Probability

The condition of two events is explained with the help of the multiplication rule probability. Assume two events, J and K, that are associated with a sample space S. When both the J and K events occur, and it is denoted by the set  J ∩ K. Hence, the occurrence of both the events J and K are represented by  ( J ∩ K ). This event can also be written as JK and using the properties of conditional probability, and you can obtain the probability of event JK. In this article, you will also learn the multiplication theorem of probability.

Proof of Multiplication Rule Probability 

Let us now learn the multiplication law of probability.

We know that the conditional probability of an event J given the fact that K has occurred is represented by P ( J | K ), and the formula goes by: 

P ( J | K ) = P(J⋂K) / P(K)

Where P ( K ) ≠ 0

P ( J ∩ K ) = P ( K ) × P ( J | K ) ……………………………………..( A )

P ( K | J ) = P ( K ∩ J ) / P ( J )

Where P ( J ) ≠ 0.

P ( K ∩ J ) = P ( J ) × P ( K | J )

Since, P ( J ∩ K ) = P ( K ∩ J )

P (  J ∩ K ) = P ( J ) × P ( K | J ) ………………………………………( B )

From (1) Jnd (2), we get:

P ( J ∩ K ) = P ( K ) × P ( J | K ) = P ( J ) × P ( K | J ) where,

P ( J ) ≠ 0, P ( K ) ≠ 0.

The result that is obtained above is known as the multiplication rule of probability.

For independent events J and K, P( K | J ) = P ( K ). The equation (2) can be modified into,

P( J ∩ K ) = P ( K ) × P ( J )

Multiplication Theorem in Probability

Now that we have learned about the multiplication rules that are implemented in probability, such as;

P ( J ∩ K ) = P ( J ) × P ( K | J ) ; if P ( J ) ≠ 0

P ( J ∩ K ) = P ( K ) × P ( J | K ) ; if P ( K ) ≠ 0

Now, let us focus on learning about the multiplication theorems for independent events of J and K.

The probability of simultaneous occurrence of two independent events will be equal to the product of the probabilities of the events if J and K are two independent events for a random experiment. Hence,

P ( J ∩ K ) = P ( J ) . P ( K )

Now, from understanding the multiplication rule, we know that;

P ( J ∩ K ) = P ( J ) × P ( K | J )

Since J and K events are independent here, we get;

P ( K | J ) = P ( K )

On solving again, we get;

P ( J ∩ K ) = P ( J ) . P ( K )

Hence, proved.

Multiplication Theorem of Probability Examples

1. A bag contains 15 red and 5 blue balls. Without the replacement of the balls, two balls are drawn from a bag one after the other. What is the probability of picking both the balls as red?

Solution:

Let J and K represent the events that the first ball is drawn and the second ball drawn is the red balls. 

Now, we have to find P ( J ∩ K ) or P ( JK ).

P ( J ) = P ( the red balls drawn in the first draw ) = 15 / 20

Now, only 14 red balls and 5 blue balls are the ones left in the bag. The conditional probability is that of a Probability of drawing a red ball in the second draw too, where the drawing of the second ball depends on the drawing of the first ball.

Therefore, the conditional probability of K on J will be,

P ( K | J ) = 14 / 19

By using the multiplication rule of probability,

P ( J ∩ K ) = P ( J ) × P ( K | J )

P ( J ∩ K ) = (15 / 20)*(14 / 19)

               = 21 / 38

Multiplication Rule of Probability for Independent Events

Independent events are defined as those in which the outcome of one event has no impact on the outcome of another. The probability multiplication method for dependent events may be extended to independent occurrences. 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 previous theorem is reduced to P(A ∩ B) = P(A) P(B). 

That is, the likelihood of both things occurring at the same time is the product of their probabilities. When events are independent, the particular multiplication rule might be used.

Multiplication Rule of Probability for Dependent Events

Dependent events are ones in which the outcome of one event has an impact on the outcome of another. The possibility of one event is influenced by the occurrence of another. The occurrence of the first event might sometimes influence the likelihood of the second event. For example, if you draw a King from a deck of cards and do not replace it, your chances of obtaining another King are reduced.

From the theorem, we have P(A ∩ B) = P(A) P(B | A), where A and B are independent events. When dealing with dependent events, the general multiplication rule must be applied.

Specific Multiplication Rule

Calculate the combined probability of separate occurrences using the particular multiplication rule. Multiply the odds of the separate events to apply this rule. The occurrence of event A has no impact on the likelihood of event B in the case of independent occurrences. The following is the particular multiplication rule in probability notation:

P(A ∩ B) = P(A) * P(B)

Alternatively, the combined likelihood of A and B happening is equal to the probability of A happening multiplied by the probability of B happening.

General Multiplication Rule

Calculate joint probability for independent or dependent occurrences using the general multiplication rule. When dealing with dependent events, the general multiplication rule is required since it allows you to account for how the occurrence of event A impacts the chance of event B. The general multiplication rule is as follows in standard notation:

P(A ∩ B) = P(A) * P(B|A)

Alternatively, the joint chance of A and B occurring is equal to the likelihood of A happening multiplied by the conditional probability of B happening if A happens.