GCSE Maths

Mutually Exclusive Events

Edexcel

Introduction

  • Studying Mutually exclusive events are essential in probability because they help us analyze different types of real-world situations where outcomes interact in different ways.
  • We learn mutually exclusive events in probability to understand situations where two events cannot happen at the same time.

What are Mutually Exclusive Events?

  • Mutually exclusive events are events that cannot happen at the same time.
  • For example: When you roll an ordinary dice, you cannot get a 3 and an even number at the same time.
  • Two events $A$ and $B$ are Mutually exclusive if,
$$P(A \cap B) = 0$$

This means there is no overlap between the two events.

  • Mathematically,
$$P(A \cup B) = P(A) + P(B)$$

Steps To Solve The Mutually Exclusive Events

Here are the steps to solve problems in probability:

1
Identify Events
2
Use Formula
3
Calculate the Probability

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Solved Examples

Solved Example
A fair six-sided die is rolled. What is the probability of rolling a 2 or a 5?
SOLUTION
1
Identify Events:
  • Event A: Rolling a 2.
  • Event B: Rolling a 5.
2
Use Formula:
$$P(A \cup B) = P(A) + P(B)$$
3
Calculate the Probability:

Put the Values in formula,

$$\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

The Probability of rolling a 2 or a 5 is $\frac{1}{3}$

Final Answer: $\frac{1}{3}$

Solved Example
A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. If one marble is drawn at random, what is the probability that it is red or blue?
SOLUTION
1
Identify Events:
  • Event A: Drawing a red marble
  • Event B: Drawing a blue marble
2
Use Formula:
$$P(A \cup B) = P(A) + P(B)$$
3
Calculate the Probability:

Total marbles: 10 marbles

Put the Values in formula,

$$\frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2}$$

The Probability of drawing a Red or Blue marble is $\frac{1}{2}$

Final Answer: $\frac{1}{2}$

Solved Example
A card is drawn from a standard deck of 52 cards. What is the probability that the card is either a Heart or a Club?
Playing cards showing a Heart and a Club
SOLUTION
1
Identify Events:
  • Event A: Drawing a Heart: 13 Hearts
  • Event B: Drawing a Club: 13 Clubs
2
Use Formula:
$$P(A \cup B) = P(A) + P(B)$$
3
Calculate the Probability:

Total cards: 52 cards

Put the Values in formula,

$$\frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2}$$

The probability of drawing either a Heart or a Club from a deck is $\frac{1}{2}$

Final Answer: $\frac{1}{2}$

Solved Example

In a class of 30 students, each child likes different subjects, such as:

  • 12 students like Math(M)
  • 8 students like Science(S)
  • 10 students like History(H)

If a student is selected at random, what is the probability that the student likes math or science?

SOLUTION
1
Identify Events:
  • Event A: Student like Math: 12
  • Event B: Student like Science: 8
2
Use Formula:
$$P(A \cup B) = P(A) + P(B)$$
3
Calculate the Probability:

Total Students: 30

Put the Values in formula,

$$\frac{12}{30} + \frac{8}{30} = \frac{20}{30} = \frac{2}{3}$$

The probability that a randomly chosen student likes either math or science is $\frac{2}{3}$

Final Answer: $\frac{2}{3}$

Solved Example

A pizza is cut into 8 equal slices:

  • 3 slices have cheese topping(C)
  • 2 slices have pepperoni topping(P)
  • 3 slices have veggie topping(V)

If a person randomly picks one slice, what is the probability that it is cheese or pepperoni?

SOLUTION
1
Identify Events:
  • Event A: Cheese slices: 3
  • Event B: Pepperoni: 2
2
Use Formula:
$$P(A \cup B) = P(A) + P(B)$$
3
Calculate the Probability:

Total Slices: 8

Put the Values in formula,

$$\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$$

The probability of picking a cheese or pepperoni slice is $\frac{5}{8}$

Final Answer: $\frac{5}{8}$

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