CBSE Class 11 Maths Revision Notes Chapter 14 Probability

Probability explains how the chance of an event is measured using outcomes, sample spaces and events. For CBSE Class 11 Maths 2026–27, this chapter covers event types, algebra of events and the axiomatic approach to probability.

Probability is an important chapter in Class 11 Mathematics. It helps students measure the chance of an event happening in a random experiment. For example, when a coin is tossed, the result can be head or tail. When a die is rolled, the result can be any number from 1 to 6. Probability helps us study such outcomes in a systematic way.

Use these CBSE Class 11 Maths Revision Notes Chapter 14 to revise random experiment, sample space, outcome, event, types of events, algebra of events, mutually exclusive events, exhaustive events and probability formulas.

These Class 11 Maths Chapter 14 Notes are useful when you want quick revision of definitions, examples and formulas before solving Probability questions.

Key Takeaways

• Sample space: The set of all possible outcomes of a random experiment is called the sample space.
• Event: Any subset of a sample space is called an event.
• Mutually exclusive events: Two events are mutually exclusive if they cannot occur together.
• Probability: Probability of an event lies between 0 and 1.

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Class 11 Maths Chapter 14 Notes for Probability Revision

These notes are arranged for 30-minute revision, so students can revise the chapter in a simple order.

Start with random experiments, outcomes and sample space. Then revise events and their types. After that, study algebra of events, mutually exclusive events, exhaustive events and the axiomatic approach to probability.

What is Probability?

Probability is the measure of the chance that an event will occur.

It is written as:

P(E)

Here, E is an event.

Probability always lies between 0 and 1.

0 ≤ P(E) ≤ 1

If P(E) = 0, the event is impossible.

If P(E) = 1, the event is sure.

If P(E) is close to 1, the event is more likely to happen.

Random Experiment

A random experiment is an experiment whose result cannot be predicted with certainty before it is performed.

Examples:

• Tossing a coin
• Rolling a die
• Drawing a card from a deck
• Tossing two coins
• Selecting a student from a class

In each case, we know the possible outcomes, but we do not know the exact result in advance.

This idea is the starting point of Class 11 Maths Probability Notes.

Outcome

An outcome is a possible result of a random experiment.

Example:

When a coin is tossed, the possible outcomes are:

H, T

Here, H means head and T means tail.

When a die is rolled, the possible outcomes are:

1, 2, 3, 4, 5, 6

Each possible result is called an outcome.

Sample Space

The sample space is the set of all possible outcomes of a random experiment.

It is usually denoted by S.

Example 1:

For tossing a coin:

S = {H, T}

Example 2:

For rolling a die:

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

Example 3:

For tossing two coins:

S = {HH, HT, TH, TT}

A sample space acts like the universal set for the experiment.

Event in Probability

An event is any subset of the sample space.

If S is the sample space, then any subset E of S is called an event.

Example:

For rolling a die:

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

Let E be the event “getting an even number”.

Then:

E = {2, 4, 6}

Since E is a subset of S, E is an event.

Occurrence of an Event

An event occurs if the actual outcome of the experiment belongs to that event.

Example:

Let E be the event “getting a number less than 4” when a die is rolled.

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

E = {1, 2, 3}

If the die shows 1, 2 or 3, event E occurs.

If the die shows 4, 5 or 6, event E does not occur.

Types of Events in Probability

Events can be classified based on the number of outcomes they contain and how they relate to other events.

Probability Class 11 Notes often test these definitions through coins, dice and cards.

Impossible Event

An impossible event is an event that cannot occur.

It is represented by the empty set:

φ

Example:

When a die is rolled, let E be the event “getting a multiple of 7”.

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

There is no multiple of 7 in S.

So:

E = φ

This is an impossible event.

Sure Event

A sure event is an event that always occurs.

It is equal to the sample space S.

Example:

When a die is rolled, let F be the event “getting an odd or even number”.

F = {1, 2, 3, 4, 5, 6}

So:

F = S

This is a sure event.

Simple Event

A simple event has exactly one sample point.

Example:

When a die is rolled, the event “getting 4” is:

E = {4}

This event has only one outcome.

So, it is a simple event.

Example with two coins:

S = {HH, HT, TH, TT}

The simple events are:

{HH}, {HT}, {TH}, {TT}

Compound Event

A compound event has more than one sample point.

Example:

When a die is rolled, the event “getting an even number” is:

E = {2, 4, 6}

This event has three outcomes.

So, it is a compound event.

Example with three coins:

Event “exactly one head appears”:

E = {HTT, THT, TTH}

This is also a compound event.

Complementary Event

For every event A, there is another event called the complementary event.

It is the event “not A”.

It is written as:

A′ or Aᶜ

If S is the sample space, then:

A′ = S - A

Example:

When a die is rolled:

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

Let A be the event “getting a prime number”.

A = {2, 3, 5}

Then:

A′ = {1, 4, 6}

So, A′ is the event “not getting a prime number”.

Algebra of Events

Events are subsets of sample space, so they can be combined using set operations.

These operations help students solve probability questions involving two or more events.

Event A or B

The event “A or B” means A occurs, B occurs, or both occur.

It is written as:

A ∪ B

Example:

When a die is rolled:

A = {2, 3, 5}, event of getting a prime number
B = {1, 3, 5}, event of getting an odd number

Then:

A ∪ B = {1, 2, 3, 5}

Event A and B

The event “A and B” means both A and B occur.

It is written as:

A ∩ B

Example:

A = {2, 3, 5}
B = {1, 3, 5}

Then:

A ∩ B = {3, 5}

These are the outcomes common to both A and B.

Event A but Not B

The event “A but not B” means outcomes that are in A but not in B.

It is written as:

A - B

Also:

A - B = A ∩ B′

Example:

A = {2, 3, 5}
B = {1, 3, 5}

Then:

A - B = {2}

Mutually Exclusive Events

Two events A and B are mutually exclusive if they cannot occur at the same time.

This means:

A ∩ B = φ

Example:

When a die is rolled:

A = event of getting an odd number = {1, 3, 5}
B = event of getting an even number = {2, 4, 6}

A ∩ B = φ

So, A and B are mutually exclusive events.

Events That Are Not Mutually Exclusive

Two events are not mutually exclusive if they can occur together.

Example:

When a die is rolled:

A = event of getting an odd number = {1, 3, 5}
B = event of getting a number less than 4 = {1, 2, 3}

A ∩ B = {1, 3}

Since A and B have common outcomes, they are not mutually exclusive.

Exhaustive Events

Events are exhaustive if at least one of them must occur whenever the experiment is performed.

If E1, E2, ..., En are events of sample space S, then they are exhaustive if:

E1 ∪ E2 ∪ ... ∪ En = S

Example:

When a die is rolled:

A = {1, 2, 3}
B = {3, 4}
C = {5, 6}

A ∪ B ∪ C = {1, 2, 3, 4, 5, 6} = S

So, A, B and C are exhaustive events.

Mutually Exclusive and Exhaustive Events

Events are mutually exclusive and exhaustive when:

• No two events occur together.
• Their union is the complete sample space.

Example:

When a coin is tossed:

A = {H}
B = {T}

A ∩ B = φ

A ∪ B = {H, T} = S

So, A and B are mutually exclusive and exhaustive events.

Axiomatic Approach to Probability

The axiomatic approach to probability defines probability using rules or axioms.

Let S be the sample space of a random experiment. Probability P is a function that assigns each event a number between 0 and 1.

The axioms are:

From these axioms:

P(φ) = 0

This means the probability of an impossible event is 0.

Probability of an Event

If all outcomes are equally likely, then:

P(E) = Number of favourable outcomes / Total number of outcomes

This is the basic probability formula.

Example:

When a die is rolled, find the probability of getting an even number.

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

E = {2, 4, 6}

Number of favourable outcomes = 3
Total outcomes = 6

P(E) = 3/6 = 1/2

Equally Likely Outcomes

Outcomes are equally likely when each outcome has the same chance of occurring.

Examples:

• Head and tail in a fair coin toss
• Numbers 1 to 6 in a fair die roll
• Any card drawn from a well-shuffled deck

If S has n equally likely outcomes, then each outcome has probability:

1/n

Example:

For a fair die:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

Probability of Complementary Event

If A is an event, then:

P(A′) = 1 - P(A)

Also:

P(A) + P(A′) = 1

Example:

When a die is rolled, let A be the event “getting an even number”.

A = {2, 4, 6}

P(A) = 3/6 = 1/2

So:

P(A′) = 1 - 1/2 = 1/2

A′ is the event “not getting an even number”.

Addition Rule of Probability

For any two events A and B:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

This is used when events may overlap.

If A and B are mutually exclusive, then:

A ∩ B = φ

So:

P(A ∪ B) = P(A) + P(B)

Example:

When a die is rolled:

A = {2, 4, 6}, event of getting an even number
B = {3, 6}, event of getting a multiple of 3

A ∩ B = {6}

P(A) = 3/6
P(B) = 2/6
P(A ∩ B) = 1/6

P(A ∪ B) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3

Probability Rules to Remember

Solved Examples on Probability

Example 1: Tossing Two Coins

A coin is tossed two times. Find the probability of getting exactly one head.

Solution:

Sample space:

S = {HH, HT, TH, TT}

Event E = exactly one head

E = {HT, TH}

n(E) = 2
n(S) = 4

P(E) = n(E)/n(S)

= 2/4

= 1/2

So, the probability of getting exactly one head is 1/2.

Example 2: Rolling a Die

A die is rolled once. Find the probability of getting a prime number.

Solution:

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

Prime numbers on a die are:

E = {2, 3, 5}

n(E) = 3
n(S) = 6

P(E) = 3/6 = 1/2

So, the probability of getting a prime number is 1/2.

Example 3: Complement Rule

A die is rolled once. Find the probability of not getting 6.

Solution:

Let A be the event “getting 6”.

A = {6}

P(A) = 1/6

Required event = not getting 6 = A′

P(A′) = 1 - P(A)

= 1 - 1/6

= 5/6

So, the probability of not getting 6 is 5/6.

Example 4: Addition Rule

A card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is a king or a red card.

Solution:

Let A be the event “getting a king”.

n(A) = 4

Let B be the event “getting a red card”.

n(B) = 26

There are 2 red kings.

n(A ∩ B) = 2

Using:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = 4/52 + 26/52 - 2/52

= 28/52

= 7/13

So, the probability is 7/13.

Example 5: Mutually Exclusive Events

A die is rolled once. Find whether the events “getting 2” and “getting 5” are mutually exclusive.

Solution:

A = {2}

B = {5}

A ∩ B = φ

Both events cannot occur in the same die roll.

So, they are mutually exclusive events.

Common Mistakes in Probability

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Important Terms from CBSE Class 11 Maths Revision Notes Chapter 14

The terms below cover the main definitions students need while revising this chapter.

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