What is Multiplication Theorem on Probability?

Multiplication theorem states that the probability of both events A and B to occur together (P(A∩B)) is determined by multiplying the probability of one event by conditional probability of the second event. Do note that we are considering that the first event has already occurred. This is an important topic for NEET exam and JEE Main exam. 

Mathematically, P(A∩B)=P(A)×P(B∣A) or P(A∩B)=P(B)×P(A∣B)

We use this rule to work out the AND probability. Like, what is the chance of A and B both happening. There are two cases: independent events and dependent events. 

1. Independent Events

Let us first consider the independent events. These events do not affect the other. The formula used for independent events is: 

P(A and B) = P(A) × P(B)

Let us first flip a coin (P = 1/2)

Then let us roll a dice (P= 1/6)

Here, chance of getting heads and a six will be = 1/2 x 1/6 = 1/12

For three or more independent event, the multiplication theorem will be P(A and B) = P(A) × P(B) × P(B).

2. Dependent Events

These are the events that do not impact one another. The formula used in case of independent events is P(A and B) = P(A) × P(B given A). Let us consider an example. You need to pick one card from a deck. After that, you pick another card without putting back the first one. In this case, the second draw depends on the first one.

Let us consider a sample space S that has events A and B. The definition of conditional probability states that: $$P(B|A)=\frac{P(A\cap B)}{P(A)}$$ provided P(A) > 0

We need to prove that P(A∩B)=P(A)⋅P(B∣A). As per conditional probability: Now, we will multiply both sides Simplifying the right-hand side. Then, we will rearrange the equation to obtain the multiplication theorem.

For independent events A and B: P(B∣A)=P(B)

Let us substitute this in theorem P(A∩B)=P(A)⋅P(B)

Let us generallize the theorem to n number of events for $$A_1,A_2,\cdots ,A_n$$ which will result in: $$P(\underset{i=1}{\overset{n}{\cap }}{A}_i)=P(A_1)\cdot P(A_2|A_1)\cdot P(A_3|A_1\cap A_2)\cdots P(A_n|\underset{i=1}{\overset{n-1}{\cap }}{A}_i)$$

Let us take a look at a special case of the multiplication theorem which is important for IIT JAM exam and CUET exam. Two events A and B will be independent events if P(B∣A)=P(B) and P(A∣B)=P(A). Based on this definition, after knowing that A has occurred dows not change probability of B.

P(B∣A)= P(B) and P(A∣B)= P(A)

For independent events, the probability of both A and B occurring together is simply product of their individual probabilities:

P(B∣A)=P(B)andP(A∣B)=P(A)

Since A and B are independet, P(B|A) = P(B) 

By substituting into general multiplication rule:

P(A∩B)=P(A)×P(B∣A)=P(A)×P(B)

In the case of independent events, you do not need to worry about conditional probabilities. You are only required to multiply individual probabilities/