Addition Rule for Probability Formula | Solved Examples, FAQs, Important Questions

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

📘 Introduction

The Addition Rule for Probability is a fundamental concept in probability theory used to calculate the probability of the occurrence of at least one of two events. It helps when two or more events can happen at the same time or independently. This rule is essential for solving problems related to real-life scenarios, board exams, and competitive exams like JEE Mains & Advanced.

Addition Rule for Probability

Mathematically probability can be defined as :

Probability of Event P(E) = [Number of Favorable Outcomes] / [Total Number of Outcomes]

The addition rule for probability is a principle that allows you to calculate the probability that at least one of two events will occur.

It is defined as the sum of the probabilities of each event, minus the probability that both events occur together. This prevents double-counting the overlap between the events.

The General Addition Rule for Probability is given by 

P(A or B) = P(A) + P(B) – P(A and B)

where A and B are the two events. 

For mutually exclusive events, P(A and B) = 0.

So P(A or B) = P(A) + P(B) for mutually exclusive events.

Mutually Exclusive Events

Two events, A and B, are said to be mutually exclusive if they cannot occur simultaneously during a single trial.

Example: In a coin toss, the events “Getting Heads” and “Getting Tails” are mutually exclusive because both cannot happen at the same time.

Explanation: “Getting Heads” and “Getting Tails” are mutually exclusive events because both cannot occur simultaneously during a single coin toss.

Addition Rule: Since P(A ∩ B) = 0 (no overlap), the formula simplifies to: P(A ∪ B) = P(A) + P(B)
Here:

• P(Getting Heads) = 1/2,
• P(Getting Tails) = 1/2,
• P(Getting Heads or Tails) = 1/2 + 1/2 = 1

When the probabilities of all possible events in a sample space are added, their sum is equal to 1.

Non-Mutually Exclusive Events

Two events, A and B, are said to be non-mutually exclusive if they can occur simultaneously during a single trial.

Example: Rolling a die, let A represent rolling an odd number ({1, 3, 5}) and B represent rolling a 3 ({3}). In this case, the number 3 belongs to both events, meaning A and B overlap.

Explanation: The outcomes 1, 5, and 3 denotes event A. 3 denotes event B. 3 is common to both the events, and thus 3 lies in the intersection. 4 and 6 do not come in any event, and thus they lie outside into the sample space. 

Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Here:

• P(A): Probability of rolling an odd number = 3/6 = 1/2,
• P(B): Probability of rolling a 3 = 1/6​,
• P(A ∩ B): Probability of rolling a number that is both odd and 3 = 1/6.

Substitute into the formula: P(A ∪ B) = 1/2 + 1/6 − 1/6 = 1/2.

Adding Probabilities 

In probability theory, when you add the probabilities of all possible outcomes of an experiment, the sum always equals 1. This is a result of the fact that one of the possible outcomes must occur.

For mutually exclusive events: The Sum of all the probabilities of all the events in an experiment is always 1.
For example: If a trial has three possible outcomes, A, B, and C. 

P(A) + P(B) + P(C) = 1

Sometimes we have only one outcome in which we are interested. Let’s say 8 teams are participating in the cricket World Cup. We are interested in finding the probability of winning the World Cup for India. We are not interested in finding out the probability for every other team. So we will formulate the problem in the following way, 

Let’s say event A denotes India winning the World Cup. So, another event B denotes India not winning the World Cup. 

P(A) + P(B) = 1

P(A) = 1 – P(B) 

Such events are called elementary events. 

Addition Rules For Probability

Suppose there are two events A and B, based on the fact whether both the events are Mutually Exclusive or not, Two different Rules are described,

Rule 1: When the events are Mutually Exclusive

When the events are mutually exclusive, the probability of the events occurring is the sum of both events.

P(A∪ B) = P(A) + P(B)

Rule 2: When the events are not mutually exclusive

There is always some overlapping between two non-mutually exclusive events, Therefore, the Probability of the events will become,

P(A∪ B) = P(A) + P(B) – P(A∩ B)

Solved Examples on Addition Rule for Probability

Question 1: Let’s say a die was rolled. Answer the following questions:

1. What is the probability of getting a number greater than 4?? 
2. What is the probability of getting an even number?

Solution: 

When a die is rolled, there are six possible outcomes: 1, 2, 3, 4, 5 and 6

1. Probability of getting a number greater than 4: 
Number of favorable outcomes = 2 
Total number of outcomes = 6

P(Number Greater than 4) = 2/6 = 1/3 

2. Probability of getting an even number: 
Number of favorable outcomes = 3 
Total number of outcomes = 6

P(Getting a Even Number) = 3/6 = 1/2

Question 2: Let’s say a card was drawn from a well-shuffled deck of cards. Find the probability of getting a Queen on one draw. 

Solution: 

We know that a deck has 52 cards. So there are total 52 outcomes that are possible if a card is drawn. We also know that there are four queens in the deck. These are our favorable outcomes. 
So, 
Total number of outcomes = 52 
Total number of favorable outcomes = 4 

P(Getting a Queeen) = 4/42 = 1/13

Question 3: A bag contains 3 white balls, 4 black balls, and 2 green balls. A ball is drawn with replacement. Find the probability of getting: 

1. A White Ball 
2. A Black Ball 
3. A Green Ball 

Solution: 

There are a total of 3 + 4 + 2 = 9 balls. 

1. Probability of getting a white ball 
Total number of balls = 9, 
Favorable outcomes = 3

P(Getting a White Ball) =3/9 = 1/3

2. Probability of getting a Black ball 
Total number of balls = 9, 
Favorable outcomes = 4

P(Getting a Black Ball) = 4/9

3. Probability of getting a Black ball 
Total number of balls = 9, 
Favorable outcomes = 2

P(Getting a Green Ball) = 2/9

Question 4: Let’s say we have a well-shuffled deck. We draw two cards and find the probability of getting either a King or a Queen. 

Solution: 

Let’s say drawing a king represents an event A while drawing a queen represents an event B. We are asked for the probability for getting either King or Queen. We will use law of adding probabilities here, 

Probability (King or Queen) = Probability (King) + Probability (Queen) 

We know that there are 4 Kings and 4 Queens in the deck. 
P(King) = 4/52 = 1/13
P(Queen) = 4/52 = 1/13

Thus, 
Probability (King or Queen) = 1/13 + 1/13 = 2/13

Question 5: We have an urn that contains three black balls, two blue balls, and three white balls. Find the probability of getting one black, one blue, and one white ball if we draw three times with replacement. 

Solution: 

We have a total of eight balls.
P(getting a black ball) = 3/8
P(getting a blue ball) = 2/8
P(getting a white ball) = 3/8

We will find out this probability with law of addition. 
So the total probability of getting all three colors = P(Black) + P(Blue) + P(White)
=3/8 × 2/8 × 3/8
= 18/512 

Notice that the probability sums up to one. This is in accordance with laws of probability. 

Question 6: The Union Budget is going to be announced by the government this week. The probability that it will be announced on a day is given, 

Day	Probability
Monday	1/7
Tuesday	3/7
Wednesday	1/7
Thursday	1/7
Friday	1/7

Find the probability of the budget getting announced between Monday to Wednesday. 

Solution: 

We need to use the probability addition law, 

P(Monday to Wednesday) = P(Monday) + P(Tuesday) + P(Wednesday)
P(Monday) = 1/7
P(Tuesday) = 3/7
P(Wednesday) = 1/7

P(Monday to Wednesday) = P(Monday) + P(Tuesday) + P(Wednesday)
= 1/7 + 3/7 + 1/7
= 5/7

Question 7: In a class of 90 students, 50 took Math, 25 took Physics, and 30 took both Math and Physics. Find the number of students who have taken either math or Physics.

Solution:

Since the events of choosing math and physics are non-mutually exclusive, the second rule of addition will be applied here,
P(Math ∪ Physics) = P(Math) + P(Physics) – P(Math ∩ Physics)
P(Math) = 50
P(physics) = 25
P(Math ∩ Physics) = 30
P(Math ∪ Physics) = 50 + 25 – 30
P(Math ∪ Physics) = 45 students.

More Practical Solved Examples

🔍 Example 1: Mutually Exclusive Events

Question:
A card is drawn from a standard deck of 52 playing cards. What is the probability of drawing either a King or a Queen?

Solution:
Let:

• A = Event of drawing a King ⇒P(A)=4/52
• BBB = Event of drawing a Queen ⇒P(B)=4/52

Since drawing a King and a Queen are mutually exclusive events:

\begin{array}{l} \mathbf{P(A \cup B) = P(A) + P(B)} \\ \mathbf{= \frac{4}{52} + \frac{4}{52}} \\ \mathbf{= \frac{8}{52} = \frac{2}{13}} \end{array}

Answer:
The probability of drawing either a King or a Queen is 2/13.

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🔍 Example 2: Non-Mutually Exclusive Events

Question:
A number is randomly selected from 1 to 20. What is the probability that the number is a multiple of 4 or a multiple of 5?

Solution:
Let:

• A = Event that the number is a multiple of 4 ⇒P(A)=5/20​ (Multiples: 4, 8, 12, 16, 20)
• BBB = Event that the number is a multiple of 5 ⇒P(B)=4/20(Multiples: 5, 10, 15, 20)
• A∩B = Multiples of both 4 and 5 (i.e., multiples of 20) ⇒P(A∩B)=1/20​

Since the events are not mutually exclusive:

\begin{array}{l} \mathbf{P(A \cup B) = P(A) + P(B) – P(A \cap B)} \\ \mathbf{= \frac{5}{20} + \frac{4}{20} – \frac{1}{20}} \\ \mathbf{= \frac{8}{20} = \frac{2}{5}} \end{array}

Answer:
The probability that the number is a multiple of 4 or 5 is 2/5​.

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🔍 Example 3: Real-life Scenario

Question:
In a class of 40 students, 18 like mathematics, 25 like science, and 10 like both subjects. What is the probability that a student selected at random likes either mathematics or science?

Solution:
Let:

• A = Event the student likes mathematics ⇒P(A)=18/40
• B = Event the student likes science ⇒P(B)=25/40
• A∩B = Event the student likes both ⇒P(A∩B)=10/40​

\begin{array}{l} \mathbf{P(A \cup B) = P(A) + P(B) – P(A \cap B)} \\ \mathbf{= \frac{18}{40} + \frac{25}{40} – \frac{10}{40}} \\ \mathbf{= \frac{33}{40}} \end{array}

Answer:
The probability that a randomly selected student likes either mathematics or science is 33/40​.

FAQs on Addition Rule for Probability

What is the Addition Rule for Probability?

The Addition Rule states that the probability of the union of two mutually exclusive events is the sum of their individual probabilities

Can the Addition Rule be applied to non-mutually exclusive events?

Yes, the Addition Rule can be extended to non-mutually exclusive events by subtracting the probability of their intersection to avoid double counting.

How is the Addition Rule related to the OR operation in probability?

In probability, the OR operation represents the union of events, which is exactly what the Addition Rule calculates: the probability of either one event or another (or both) occurring.

Can the Addition Rule be generalized to more than two events?

Yes, the Addition Rule can be extended to any finite number of events by iteratively applying it to pairs of events until all events are accounted for. However, for non-mutually exclusive events, adjustments may be needed to avoid over-counting probabilities.

✅ Key Points to Remember

• Use the basic addition rule when events are mutually exclusive.
• Subtract P(A∩B) when events can occur simultaneously (non-mutually exclusive).
• The sum of probabilities of all mutually exclusive events in a sample space is 1.

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🎯 Applications of Addition Rule in Real Life

• Calculating the probability of drawing specific cards from a deck.
• Determining the chances of weather events like rain or snow on the same day.
• Finding the likelihood of winning different prizes in a lottery.

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