How to Find Conditional Probability for Disjoint Events?

The probability of an event is a measure of the chance of occurrence of an event when an experiment is done. Two events that do not occur at the same time are called disjoint events. These are also known as mutually exclusive events. For example, consider the experiment of rolling a die, a sample space is S = {1, 2, 3, 4, 5, 6}. Let A be an event that an odd number appears and B be an event an even number appears. Here A ⋂ B = φ. A and B are disjoint sets.

In this article, we will learn how to find conditional probability for disjoint events.

What is Conditional Probability?

Conditional probability is the probability of one event occurring with some relation to one or more other events. It is denoted by P(A|B). It means the probability of A given that B has already occurred.

Disjoint Events

Two events that do not occur at the same time are called disjoint events.

Formula

P(A|B) = Number of elementary events favourable to A ⋂ B / Number of elementary events which are favourable to B.

P(A|B) = P (A ⋂ B) / P(B)

When A and B are disjoint events, P (A ⋂ B) = 0

So P(A|B) = 0.

Let A and B are any two events of a sample space S and F is an event of S such that P(F) ≠ 0

then P((A⋃B) |F) = P(A|F) + P(B|F) – P((A ⋂ B)|F).

If A and B are disjoint events, then P((A⋃B) |F) = P(A|F) + P(B|F).

Example

Let us understand with the help of an example how to find the conditional probability for disjoint events.

Find the probability that a single toss of a die will result in a number less than 4 if it is given that the toss resulted in an odd number.

Solution:

Let event A be the event that the toss resulted in an odd number and event B be the event that the number is less than 4.

A = { 1,3,5}

B = { 4}

A ⋂ B = {1,3}

P(A ⋂ B) = ⅓

P(B |A) = P(A⋂B)/P(A) = (1/3)/(1/2) = 2/3

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