Probability Class 11 Maths Notes - Chapter 16

According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 14.

Probability is an integral part of the class 11 maths syllabus and is important for class 11 exams and different engineering exams like JEE. In earlier classes, you may have studied the concept of probability as a measure of uncertainty of various phenomena. Here, a brief introduction to probability is given based on the class 11 maths syllabus, which will help students learn the related concepts quickly and score good marks in the exam. Also, learn probability and statistics here.

Probability Class 11 Chapter 16 Concepts

The topics and subtopics covered in Class 11 Probability are listed below:

• Introduction
• Random Experiments
  Outcomes and sample space

• Event
  Occurrence of an event
  Types of events
  Complementary Event
  Mutually exclusive events
  Exhaustive events

• Axiomatic Approach to Probability
  Probability of an event
  Probabilities of equally likely outcomes

For more information on Probability, watch the below video.

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Probability for Class 11: Key Concepts

• An experiment is said to be a random experiment if there is more than one possible outcome, and it is impossible to predict the outcome in advance.
• All possible results of an experiment are called its outcomes.
• Let us consider an experiment of rolling a die. All possible outcomes are 6, 5, 4, 3, 2, or 1. The set of all these outcomes {6, 5, 4, 3, 2, 1} is known as the sample space and is denoted by ‘S’.
• Let us consider an experiment of tossing Two coins once. Since the coin can turn up Tail or Head, therefore, all the possible outcomes are:
  Both coins – Head = HH, Both coins – Tail = TT, First coin – Head and Second coin – Tail = HT, First coin – Tail and Second coin – Head = TH.
  Thus, the sample space (S) can be represented as {HH, TT, HT, TH}.
• For any random experiment, let S be the sample space. The probability P is a real-valued function whose domain is the power set of S and [0, 1] is the range interval.
• For any event E, P(E) ≥ 0
• P(S) = 1
• If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F).

Events in Probability

As defined above, the set of all possible outcomes is known as the Sample space. All elements of a sample space are known as Sample points. An event is a subset of the S (sample space). An empty set is also known as the Impossible event. Event A′ is known as Complementary event A.

To know more about Events and Types of Events in Probability, visit here.

All outcomes with equal probability are called equally likely outcomes. Let S be a finite sample space with equally likely outcomes and A be the event. Therefore, the probability of event A is:
P(A) = n(A)/n(S)
Where n(A) = Number of elements on the set A
n(S) = Total number of outcomes or the number of elements in the sample space S

Let P and Q be any two events, then the following formulas can be derived.

• Event P or Q: The set P ∪ Q
• Event P and Q: The set P ∩ Q
• Event P and not Q: The set P – Q
• P and Q are mutually exclusive if P ∩ Q = φ
• Events P₁, P₂, . . . . . , P_n are exhaustive and mutually exclusive if P₁ ∪ P₂ ∪ . . . . . ∪ P_n = S and E_i ∩ E_j = φ for all i ≠ j.

To learn more about the algebra of events, visit here.

Axiomatic Approach to Probability

The axiomatic approach is a different way of defining the probability of an event. In this method, some axioms or rules are depicted to indicate possibilities. 

Let S be the sample space of a random experiment. 

The probability P is a real-valued function such that the domain of P is the power set of S and range is the interval [0, 1] satisfying the following axioms: 

(i) For any event E, P(E) ≥ 0 

(ii) P (S) = 1 

(iii) If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F). 

It follows from (iii) that P(φ) = 0. 

This can be proved as given below:

Let F = φ and note that E and φ are disjoint events. 

Therefore, from axiom (iii), we get; 

P (E ∪ φ) = P (E) + P (φ) or P(E) = P(E) + P (φ) i.e. P (φ) = 0

Let ω₁, ω₂, …, ω_n be the outcomes of a sample space S. 

i.e., S = {ω₁, ω₂, …, ω_n} 

It follows from the axiomatic definition of probability that 

(i) 0 ≤ P (ω_i ) ≤ 1 for each ω_i ∈ S 

(ii) P (ω₁ ) + P (ω₂ ) + … + P (ω_n ) = 1 

(iii) For any event A, P(A) = ∑ P(ω_i ), ω_i ∈ A.

Visit here to learn more about axiomatic probability.

Probability of an event: For a finite sample space with equally likely outcomes 

Probability of an event P(A) = n(A)/n(S)

where 

n(A) = number of elements in the set A

n(S) = number of elements in the set S

If P and Q are two events, then P(P or Q) = P(P) + P(Q) – P(P and Q), i.e. P(P ∪ Q) = P(P) + P(Q) – P(P ∩ Q)

• If P and Q are mutually exclusive, then P(P or Q) = P(P) + P(Q)
• If M is an event, then P(not M) = 1 – P(M)
• P(sure event) = 1
• P(impossible event) = 0

Solved Examples

Example 1: 

A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

Solution:

Let R denotes the red ball, and W denotes the white ball. 

Given that a box contains 1 red and 3 identical white balls.

To draw two balls at random in succession without replacement, the sample space can be written as:

S = {RW, WR, WW}

= {(R,W), (W,R),(W,W)}

Example 2:

An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: 

A: the sum is greater than 8, 

B: 2 occurs on either die 

C: the sum is at least 7 and a multiple of 3. 

Which pairs of these events are mutually exclusive?

Solution:

Given that a pair of dice rolled.

Sample space = 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)}

n(S) = 36

Event A: The sum is greater than 8

A = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}

Event B: 2 occurs on either die

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

Event C: The sum is at least 7 and a multiple of 3

C = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}

Here,

A ∩ B = Φ 

B ∩ C = Φ

A ∩ B ≠ Φ

Therefore, the pair of events A, B and, B, C are mutually exclusive.

Probability Class 11 Practice Questions

Q.1: Two coins (a one-rupee coin and a two-rupee coin) are tossed once. Find a sample space.

Q.2: Find the sample space associated with the experiment of rolling a pair of dice (one is blue and the other red) once. Also, find the number of elements of this sample space.

Q.3: In each of the following experiments, specify the appropriate sample space.
(i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his pocket. He takes two coins out of his pocket, one after the other.
(ii) A person is noting down the number of accidents along a busy highway during a year.

Q.4: A coin is tossed. If it shows the head, we draw a ball from a bag consisting of 3 blue and 4 white balls; if it shows the tail we throw a die. Describe the sample space of this experiment.

Q.5: Consider the experiment in which a coin is tossed repeatedly until a head comes up. Describe the sample space.

Q.6: Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, and B is the event ‘getting an odd number’. Write the sets representing the events (i) A or B (ii) A and B (iii) A but not B (iv) ‘not A’.

Q.7: Two dice are thrown, and the sum of the numbers which come upon the dice is noted. Let us consider the following events associated with this experiment
A: ‘The sum is even’.
B: ‘The sum is a multiple of 3’.
C: ‘The sum is less than 4’.
D: ‘The sum is greater than 11’.
Which pairs of these events are mutually exclusive?

Q.8: A coin is tossed three times. Consider the following events.
A: ‘No head appears’,
B: ‘Exactly one head appears,’ and

C: ‘At least two heads appear’.
Do they form a set of mutually exclusive and exhaustive events?

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