Probability

CBSE Class 11 Maths · 10 questions · 24 marks

This chapter reframes chance as arithmetic: every random experiment has a sample space of equally likely outcomes, and the probability of an event is simply how large a slice of that space it occupies. Building on your set theory from Chapter 1, you will compute probabilities of unions and complements using the same Venn-diagram logic, applied now to coins, dice, and cards instead of abstract sets. This axiomatic approach is the foundation for all probability and statistics you will study later.

Quick revision

A random experiment's sample space S is the set of all possible outcomes; an event is any subset of S.
Classical probability: P(E) = n(E)/n(S), the number of favourable outcomes divided by the total number of equally likely outcomes.
0 ≤ P(E) ≤ 1 for any event E; P(S) = 1 and P(∅) = 0 (impossible event).
P(E') = 1 - P(E), where E' is the complement of E (E does not occur).
Addition theorem: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).
For mutually exclusive and exhaustive events A₁, A₂, ..., Aₙ partitioning S: P(A₁)+P(A₂)+...+P(Aₙ) = 1.
When tossing two dice, n(S) = 36; when tossing a coin n times, n(S) = 2ⁿ; a standard deck has 52 cards, 4 suits of 13 each.
P(A ∩ B') = P(A) - P(A ∩ B), the probability that A occurs but B does not.

Important questions

1
MCQ1 mark
A die is thrown once. The probability of getting a number greater than 4 is:
1. A.1/6
2. B.1/3
3. C.1/2
4. D.2/3
Answer
The probability is 1/3. Sample space S = {1,2,3,4,5,6}, so n(S) = 6. Numbers greater than 4 are {5,6}, so n(E) = 2. P(E) = 2/6 = 1/3.
2
MCQ1 mark
Two coins are tossed simultaneously. The probability of getting at least one head is:
1. A.1/4
2. B.1/2
3. C.3/4
4. D.1
Answer
The probability is 3/4. Sample space S = {HH, HT, TH, TT}, so n(S) = 4. At least one head means all outcomes except TT, so n(E) = 3. P(E) = 3/4.
3
MCQ1 mark
If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, then P(A ∪ B) equals:
1. A.0.6
2. B.0.7
3. C.0.9
4. D.1.1
Answer
P(A ∪ B) = 0.7. Using the addition theorem: P(A∪B) = P(A) + P(B) - P(A∩B) = 0.4 + 0.5 - 0.2 = 0.7.
4
MCQ1 mark
A card is drawn at random from a well-shuffled deck of 52 cards. The probability that it is a king or a queen is:
1. A.1/13
2. B.2/13
3. C.4/13
4. D.1/26
Answer
The probability is 2/13. There are 4 kings and 4 queens, so 8 favourable cards out of 52 (kings and queens cannot overlap, so they're mutually exclusive). P = 8/52 = 2/13.
5
Assertion & Reason1 mark
Assertion (A): If P(A) = 0.6, then P(not A) = 0.4. Reason (R): For any event A, P(A) + P(A') = 1.
1. A.Both A and R are true and R is the correct explanation of A
2. B.Both A and R are true but R is not the correct explanation of A
3. C.A is true but R is false
4. D.A is false but R is true
Answer
Both A and R are true and R is the correct explanation of A. From P(A)+P(A')=1 with P(A)=0.6, we get P(A')=1-0.6=0.4, matching Assertion A, which is true. Reason R states exactly the complement rule used to derive this value, so R correctly explains A.
6
Short answer2 marks
A bag contains 5 red balls and 7 black balls. One ball is drawn at random. Find the probability that it is red.
7
Short answer3 marks
Two dice are thrown together. Find the probability that the sum of the numbers appearing is 8.
8
Long answer5 marks
In a class of 60 students, 30 study Physics, 25 study Chemistry, and 12 study both. A student is selected at random. Find the probability that the student studies (i) Physics or Chemistry, (ii) neither subject, (iii) exactly one of the two subjects.
9
Long answer5 marks
A committee of 4 people is to be chosen at random from a group of 7 men and 5 women. Find the probability that the committee has (i) exactly 2 women, (ii) at least 3 women.
10
Case study4 marks
Read the following and answer the questions that follow: A survey of 100 office workers found that 55 drink tea, 40 drink coffee, and 20 drink both tea and coffee. One worker is selected at random from this group. (a) Find the probability that the worker drinks tea. (b) Find the probability that the worker drinks tea or coffee. (c) Find the probability that the worker drinks neither tea nor coffee. (d) Find the probability that the worker drinks exactly one of the two drinks.

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Probability

CBSE Class 11 Maths · 10 questions · 24 marks

Quick revision

A random experiment's sample space S is the set of all possible outcomes; an event is any subset of S.

Classical probability: P(E) = n(E)/n(S), the number of favourable outcomes divided by the total number of equally likely outcomes.

0 ≤ P(E) ≤ 1 for any event E; P(S) = 1 and P(∅) = 0 (impossible event).

P(E') = 1 - P(E), where E' is the complement of E (E does not occur).

Addition theorem: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

For mutually exclusive and exhaustive events A₁, A₂, ..., Aₙ partitioning S: P(A₁)+P(A₂)+...+P(Aₙ) = 1.

When tossing two dice, n(S) = 36; when tossing a coin n times, n(S) = 2ⁿ; a standard deck has 52 cards, 4 suits of 13 each.

P(A ∩ B') = P(A) - P(A ∩ B), the probability that A occurs but B does not.

Important questions

MCQ1 mark

A die is thrown once. The probability of getting a number greater than 4 is:

A.1/6
B.1/3
C.1/2
D.2/3

MCQ1 mark

Two coins are tossed simultaneously. The probability of getting at least one head is:

A.1/4
B.1/2
C.3/4
D.1

MCQ1 mark

If P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, then P(A ∪ B) equals:

A.0.6
B.0.7
C.0.9
D.1.1

MCQ1 mark

A card is drawn at random from a well-shuffled deck of 52 cards. The probability that it is a king or a queen is:

A.1/13
B.2/13
C.4/13
D.1/26

Assertion & Reason1 mark

Assertion (A): If P(A) = 0.6, then P(not A) = 0.4. Reason (R): For any event A, P(A) + P(A') = 1.

A.Both A and R are true and R is the correct explanation of A
B.Both A and R are true but R is not the correct explanation of A
C.A is true but R is false
D.A is false but R is true

Short answer2 marks

A bag contains 5 red balls and 7 black balls. One ball is drawn at random. Find the probability that it is red.

Short answer3 marks

Two dice are thrown together. Find the probability that the sum of the numbers appearing is 8.

Long answer5 marks

In a class of 60 students, 30 study Physics, 25 study Chemistry, and 12 study both. A student is selected at random. Find the probability that the student studies (i) Physics or Chemistry, (ii) neither subject, (iii) exactly one of the two subjects.

Long answer5 marks

A committee of 4 people is to be chosen at random from a group of 7 men and 5 women. Find the probability that the committee has (i) exactly 2 women, (ii) at least 3 women.

Case study4 marks

Read the following and answer the questions that follow: A survey of 100 office workers found that 55 drink tea, 40 drink coffee, and 20 drink both tea and coffee. One worker is selected at random from this group. (a) Find the probability that the worker drinks tea. (b) Find the probability that the worker drinks tea or coffee. (c) Find the probability that the worker drinks neither tea nor coffee. (d) Find the probability that the worker drinks exactly one of the two drinks.