# Mutually Exclusive Events: Definition, Formulas, Solved Examples

A foundational area of mathematics that addresses the possibility of events happening is called probability theory. One of its basic ideas is that of events that cannot occur simultaneously. This blog post will discuss mutually exclusive occurrences, the formulae that go along with them, and some solved instances to help you understand these ideas.

## Definition of Mutually Exclusive Events

When two events cannot happen simultaneously, they are considered mutually exclusive. Stated differently, the occurrence of one event precludes the possibility of the other.

### Examples of Mutually Exclusive Events

• Rolling a die: The outcomes of rolling a 3 and rolling a 5 are mutually exclusive since a single die roll cannot result in both 3 and 5.
• Tossing a coin: The outcomes of landing heads and landing tails are mutually exclusive because a single coin toss cannot result in both heads and tails.
• Formulas for Mutually Exclusive Events

The probability formulas associated with mutually exclusive events are straightforward due to their defining characteristic.

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### Formulas for Mutually Exclusive Events

The probability formulas associated with mutually exclusive events are straightforward due to their defining characteristic.

• Addition Rule for Mutually Exclusive Events: If A and B are mutually exclusive events, then the probability that either A or B occurs is the sum of their individual probabilities:

P(A∪B)=P(A)+P(B)

• General Addition Rule (Non-Mutually Exclusive Events for Comparison): If A and B are not mutually exclusive, the formula accounts for the overlap:

P(A∪B)=P(A)+P(B)−P(A∩B)

• Complement Rule: The probability that event A does not occur is the complement of A:

P()=1−P(A)

Solved Examples

### Example 1: Rolling a Die

Problem:
Calculate the probability of rolling either a 2 or a 4 on a fair six-sided die.

Solution:
Let A be the event of rolling a 2, and B be the event of rolling a 4. These events are mutually exclusive.

P(A)=

P(B)=

Using the addition rule for mutually exclusive events:

P(A∪B)=P(A)+P(B)

The probability of rolling either a 2 or a 4 is ​​.

### Example 2: Drawing a Card from a Deck

Problem:
Calculate the probability of drawing either a king or a queen from a standard deck of 52 cards.

Solution:
Let A be the event of drawing a king, and B be the event of drawing a queen. These events are mutually exclusive.

P(A)= ​​

P(B)=

Using the addition rule for mutually exclusive events:

P(A∪B)=P(A)+P(B)= =

The probability of drawing either a king or a queen is ​.

### Example 3: Selecting a Fruit

Problem:
In a basket with 3 apples, 2 oranges, and 5 bananas, what is the probability of selecting either an apple or an orange?

Solution:
Let A be the event of selecting an apple, and B be the event of selecting an orange. These events are mutually exclusive.

P(A)= ,

P(B)=

Using the addition rule for mutually exclusive events:

P(A∪B)=P(A)+P(B)= + ​=​=​ ​

The probability of selecting either an apple or an orange is ​​.

Comprehending occurrences that are mutually exclusive facilitates the computation of probability in many situations. The chance of either event happening may be simply ascertained by applying the addition rule. The given examples show how these ideas are used in practical settings, improving our capacity to precisely and effectively tackle probability problems.

Q1: What are Mutually Exclusive Events?

Mutually exclusive events are two or more events that cannot occur simultaneously. If one event happens, the other cannot. This concept is fundamental in probability theory.

Example: Consider the roll of a single six-sided die:

• Event A: Rolling a 3.
• Event B: Rolling a 5.

Since a die roll can only result in one number, events A and B are mutually exclusive.

Q2: What is the Formula for Mutually Exclusive Events?

Ans: Basic Probability Formula: If A and B are mutually exclusive events, the probability of either event A or event B occurring is given by: P(A∪B)=P(A)+P(B)

Generalized Formula for Multiple Events: For events A1,A2,…, ​ that are mutually exclusive: P(A1∪A2∪…∪An)=P(A1)+P(A2)+…+P(An)

Q3: How to Determine if Events are Mutually Exclusive?

Ans: Method: Check if the intersection of the events is empty. Mathematically, two events A and B are mutually exclusive if: P(A∩B)=0

Q4: What is the Difference Between Mutually Exclusive and Independent Events?

Ans: Mutually Exclusive Events:

• Cannot happen at the same time.
• Example: Rolling a die and getting a 3 or a 5.

Independent Events:

• The occurrence of one event does not affect the probability of the other.
• Example: Flipping Q coin and rolling a die.
Q5: What Happens When Events are Not Mutually Exclusive?

Ans: If events are not mutually exclusive, they can occur at the same time. For such events A and B, the probability of either A or B occurring is given by: P(A∪B)=P(A)+P(B)−P(A∩B)

Here, P(A∩B) is subtracted to avoid double-counting the outcomes that are common to both events.

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