Mutually Exclusive Events

In probability theory, two events are said to be mutually exclusive if they cannot occur at the same time or simultaneously. In other words, mutually exclusive events are called disjoint events. If two events are considered as disjoint events, then the probability of both events occurring at the same time will be zero.

If A and B are the two events, then the probability of probability is written by

Probability of Disjoint (or) Mutually Exclusive Event = P ( A and B) = 0

In probability, the specific addition rule is valid when two events are mutually exclusive. It states that the probability of either event occurring is the sum of probabilities of each event occurring.

If A and B are said to be mutually exclusive events then the probability of an event A occurring or the probability of event B occurring is given as P(A) + P(B)

P (A or B) = P(A) + P(B)

Some of the examples of the mutually exclusive events are:

• When tossing a coin, the event of getting head and tail are mutually exclusive. Because the probability of getting head and tail simultaneously is 0.
• In a six-sided die, the events “2” and “5” are mutually exclusive. We cannot get both the events 2 and 5 at the same time when we threw one die.
• In a deck of 52 cards, drawing a red card and drawing a club are mutually exclusive events because all the clubs are black.

If the events A and B are not mutually exclusive, the probability of getting A or B is given as

P (A or B) = P(A) + P(B) – P (A and B)

Dependent and Independent Events

Two events are said to be dependent if the occurrence of one event changes the probability of another event. Two events are said to be independent events if the probability of one event that does not affect the probability of another event. If two events are mutually exclusive, they are not independent and also independent events cannot be mutually exclusive.

Mutually Exclusive Events Probability Rules

From the definition of mutually exclusive events, certain rules for the probability are concluded.

Addition Rule: P ( A + B ) = 1

Subtraction Rule: P ( A U B) = 0

Multiplication Rule: P ( A ∩ B ) = 0

Conditional Probability for Mutually Exclusive Events

Conditional probability is stated as the probability of an event A, given that another event B has occurred. Conditional Probability for two independent events B given A is denoted by the expression P( B|A) and it is defined using the equation

$P(B|A)= \frac{P(A\cap B)}{P(A)}$

Redefine the above equation using multiplication rule: P ( A∩B ) = 0

$P(B|A)= \frac{0}{P(A)}$

So the conditional probability formula for mutually exclusive events is

P ( B | A) = 0

Solved Problem

Here the sample problem for mutually exclusive events is given in detail. Go through once to learn easily.

Question

What is the probability of a die showing a number 3 or number 5?

Solution:

Let,

P(3) is the probability of getting a number 3

P(5) is the probability of getting a number 5

P(3) = 1/6 and P(5) = 1/6

So,

P( 3 or 5) = P(3) + P(5)

P ( 3 or 5) = (1/6) + (1/6) = 2/6

P( 3 or 5) = 1/3

Therefore, the probability of a die showing 3 or 5 is 1/3

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Practise This Question

The argument of the complex number -1 + i 3 is
[MP PET 1994]