Probability of Compound Events: Formula, Types and Examples

Compound probability calculates the chance of two or more events happening together. There are two types of compound probability: mutually exclusive and mutually inclusive compound events. Understanding the probability of compound events helps with real-life probability problems, like when you are tossing coins, drawing cards, throwing dice, or analysing the outcomes in exams. Compound probability calculates the likelihood of multiple events happening together. In this guide, you will learn about the definition of compound probability, its properties and understand it better through simple and clear examples.

Table of Contents

• What Is the Probability of Compound Events?
• Formula for Probability of Compound Events
• Types of Probability of Compound Events
• How to Calculate the Probability of Compound Events
• Solved Examples on Probability of Compound Events
• Practice Questions on Probability of Compound Events
• Frequently Asked Questions on Probability of Compound Events

What is Probability of Compound Events?

Compound Events: A compound event is an event that involves two or more simple events occurring together or simultaneously.

Compound probability is the probability of two or more independent events occurring together. Independent events are events whose outcomes do not affect each other. The probability of compound events always lies between 0 and 1.

Formula for Probability of Compound Events

To find probability of compound events, we use different formulas depending on whether the events are independent or dependent.

Compound Event	Description	Formula
Independent Events	Independent events are those in which the outcome of one event does not affect the outcome of the other.	$P(A \cap B)=P(A)\times P(B)$
Dependent Events	Dependent events are those in which the outcome of the first event changes the probability of the second event.	$P(A \cap B)=P(A)\times P(B\mid A)$

Where:

• P(A) = Probability of Event A
• P(B) = Probability of Event B
• P(A ∩ B) = Probability of both events occurring
• P(B | A) = Probability of the second event after the first event has occurred

Types of Compound Probability

There are 3 types of compound probability:

Mutually exclusive events compound probability: When two events cannot happen at the same time, they are called mutually exclusive events.

• Let A and B be two mutually exclusive events. P(A or B) = P(A) + P(B)
• $P(A\cup B) = P(A) + P(B)$

Mutually inclusive events compound probability: When two events can happen at the same time, they are called mutually inclusive events.

• Let A and B be two mutually inclusive events. P(A or B) = P(A) + P(B) - P(A and B)
• $P(A\cup B) = P(A) + P(B) - P(A\cap B)$

Independent events: Two events are independent if the outcome of one does not affect the other. P(A and B) = P(A) × P(B)

How to Calculate the Probability of Compound Events

Many students know the formula but get confused about which formula to use. Follow this simple three-step method.

Step 1: Identify the Events

Read the question carefully and mark the events.

Example: 'A coin is tossed and a die is rolled.'

Event A = Getting a Head

Event B = Getting a 6

Step 2: Check Whether the Events Are Independent or Dependent

Examples:

Tossing a coin and rolling a die → Independent

Drawing two cards without replacement → Dependent

Step 3: Apply the Correct Formula

Solved Examples on Probability of Compound Events

Example 1: Two dice are thrown simultaneously. Find the probability that the sum is 8.

Solution: Total outcomes when two dice are thrown simultaneously = 36 

S = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1) (3,2) (3,3) (3,4) (3,5), (3,6) (4,1) (4,2) (4,3) (4,4), (4,5) (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3) (6,4) (6,5) (6,6)}

E = sum is 8 = {(2,6), (3,5), (4,4),(5,3), (6,2)}

Favourable outcomes = 5

P(E) = 5/36

Example 2: A coin is tossed, and a die is rolled. Find the probability of getting a head and a 4.

Solution: P(H) = 1/2 P(4) = 1/6

P(getting a head and a 4) = (1/2) × (1/6) = 1/12. (∵ the events are independent)

Example 3: Two cards are drawn from a deck without replacement. Find the probability that both are aces.

Solution: P(First ace) = 4/52

P(Second ace) = 3/51

P(both cards are aces) = P(first ace) × P(second ace) = 4/52 × 3/51 = 1/221

Example 4: What is the probability of drawing a card that is a heart or a king from a deck of 52 cards?

Solution: P (heart) = 13/52

P (king) = 4/52,

P(heart and king) =  $P(heart\cap king)$ = 1/52 

P(heart or king) =  $P(heart\cup king)$ = P(heart) + P(king) -  $P(heart\cap king)$ = (13/52) + (4/52) - (1/52) = 4/13 = 0.30

Practice Questions on Probability of Compound Events

1. A die is rolled twice. What is the probability of getting 6 on both rolls?
2. A card is drawn from a deck. What is the probability of getting a king or a queen?
3. A bag contains 5 red and 5 blue balls. One ball is drawn. What is the probability of getting red?
4. A die is rolled and a coin is tossed. What is the probability of getting a number greater than 4 and a tail?
5. A bag has 3 white and 2 black balls. Two balls are drawn with replacement. What is the probability of getting two white balls?
6. A bag contains 4 red, 3 blue, and 3 green balls. Two balls are drawn without replacement. Find the probability that both are blue.
7. Two dice are rolled. Find the probability that both numbers are prime.

Know more about related topics:

• Probability
• Probability Questions
• Class 7 Maths Probability
• Complementary Events
• Binomial Distribution