Probability of Compound Events

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 Compound Probability
• Types of Compound Probability
• Solved Examples on Compound Probability

What is Compound Probability

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.

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∪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∪B)=P(A)+P(B)−P(A∩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)

Solved Examples on Compound Probability

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∩king) = 1/52 

P(heart or king) =  P(heart∪king) = P(heart) + P(king) -  P(heart∩king) = (13/52) + (4/52) - (1/52) = 4/13 = 0.30