Probability Class 11 Notes & Important Questions (Ch 14)

Revision notes

Probability: revision notes

1. Random experiments and sample spaces

A random experiment has more than one possible outcome and the outcome cannot be predicted. The sample space $S$ is the set of all outcomes: three coins give $2^{3} = 8$ outcomes; two dice give $36$ ordered pairs.

2. Events

An event is a subset of $S$ . Impossible event $ϕ$ ; sure event $S$ ; simple event: one outcome.
'Not $A$ ' is $A^{'} = S - A$ ; ' $A$ or $B$ ' is $A \cup B$ ; ' $A$ and $B$ ' is $A \cap B$ ; ' $A$ but not $B$ ' is $A - B = A \cap B^{'}$ .
Mutually exclusive: $A \cap B = ϕ$ . Exhaustive: $A_{1} \cup A_{2} \cup \dots = S$ .

3. Axiomatic probability

A probability $P$ assigns to each event a number with (i) $P (E) \geq 0$ , (ii) $P (S) = 1$ , (iii) $P (E \cup F) = P (E) + P (F)$ when $E, F$ are mutually exclusive. For a finite sample space the probabilities of the outcomes are non-negative and add to $1$ . With $n$ equally likely outcomes, $P (E) = \frac{n (E)}{n (S)}$ (the classical definition).

4. Rules

$P (not A) = 1 - P (A)$ ; $0 \leq P (A) \leq 1$ ; $P (ϕ) = 0$ .
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$ ; for mutually exclusive events $P (A \cup B) = P (A) + P (B)$ .
$P (A \cap B^{'}) = P (A) - P (A \cap B)$ ; $P (neither) = 1 - P (A \cup B)$ .

5. Counting with combinations

When a selection is made at random, count favourable and total selections with $^{n} C_{r}$ : for example the probability that a random committee has exactly $2$ girls is $\frac{^{g} C_{2} \times^{b} C_{r - 2}}{^{g + b} C_{r}}$ .

Worked example 1

Two dice are thrown. Find $P (sum = 5)$ .

$(1, 4), (2, 3), (3, 2), (4, 1)$ : $\frac{4}{36} = \frac{1}{9}$ .

Worked example 2

$P (A) = 0.6$ , $P (B) = 0.5$ , $P (A \cup B) = 0.8$ . Find $P (A \cap B)$ .

$0.6 + 0.5 - 0.8 = 0.3$ .

Worked example 3

Two cards are drawn from $5$ cards numbered $1$ to $5$ . Find the probability that both are odd.

$\frac{^{3} C_{2}}{^{5} C_{2}} = \frac{3}{10}$ .

Common errors

Adding $P (A) + P (B)$ for events that can happen together (the overlap is counted twice).
Listing unordered outcomes for two dice, which are not equally likely.
Assigning probabilities that are negative or do not add up to $1$ .
Confusing "mutually exclusive" (cannot both happen) with "exhaustive" (between them cover everything).

Exam tips

Name the events with letters and write the rule you use before substituting.
Check that every answer lies between $0$ and $1$ .

Topics in this chapter: Random experiments and sample spaces · Rules · Axiomatic probability · Events · Counting with combinations.

Formulas for this chapter: Chapter 14 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can write sample spaces, describe 'not', 'and' and 'or' events and use P(not A) = 1 − P(A).

Read first: 1. Random experiments and sample spaces; 2. Events; 4. Rules (not) 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can use the addition rule, test assignments against the axioms and count with combinations.

Read first: 3. Axiomatic probability; 4. Addition rule; 5. Counting with combinations; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can solve multi-part problems with dice, cards, committees and survey data.

Read first: 4. Rules; 5. Counting 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can combine mutually exclusive cases and complements in less direct questions.

Read first: 2. Mutually exclusive events; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 40 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceRandom experiments and sample spaces

A coin is tossed three times. The number of outcomes in the sample space is

(a) $3$
(b) $6$
(c) $8$
(d) $9$

Q2·1 mark·Multiple choiceRules

If $P (A) = 0.35$ , then $P (not A)$ is

(a) $0.65$
(b) $0.35$
(c) $0.75$
(d) $1.35$

Q3·1 mark·Multiple choiceRules

If $P (A) = 0.5$ , $P (B) = 0.3$ and $P (A \cap B) = 0.1$ , then $P (A \cup B)$ is

(a) $0.8$
(b) $0.7$
(c) $0.9$
(d) $0.6$

Worked solution·2 marks·Very short answerEvents

A die is thrown. Let $A$ be “a number less than $4$ ” and $B$ be “an odd number”. Write $A \cup B$ , $A \cap B$ and $A^{'}$ .

Full worked solution Free, no sign-in

$A = {1, 2, 3}$ , $B = {1, 3, 5}$ : $A \cup B = {1, 2, 3, 5}$ A1
$A \cap B = {1, 3}$ , $A^{'} = {4, 5, 6}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S101 · Problem S29 · Problem S102 · Problem S30. All 20 →

Knowledge organiser: Probability on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Probability

P(A ∪ B) = P(A) + P(B) used when A and B overlap. Fix: subtract P(A ∩ B) unless they are mutually exclusive.
Two-dice outcomes treated as unordered. Fix: 36 ordered pairs; (1, 2) and (2, 1) are different.
Invalid probability assignment accepted. Fix: each value ≥ 0 and the total exactly 1.
'A but not B' found as P(A) − P(B). Fix: P(A) − P(A ∩ B).
Selections counted with permutations. Fix: random committees and hands are combinations.

Test Probability against the clock

Want it against the clock? Take a timed 30-mark chapter test on Probability, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on statistics and probability.

Probability: common questions

What is in CBSE Class 11 Maths Chapter 14, Probability?

The topics are: Random experiments and sample spaces, Rules, Axiomatic probability, Events, Counting with combinations. This page has revision notes for each, and 40 board-style practice questions with step-marked solutions.

How many marks is Probability worth?

It is in the Statistics and Probability unit, which carries 12 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Probability?

P(A ∪ B) = P(A) + P(B) used when A and B overlap. Fix: subtract P(A ∩ B) unless they are mutually exclusive. Two-dice outcomes treated as unordered. Fix: 36 ordered pairs; (1, 2) and (2, 1) are different.

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Probability Class 11: notes and important questions

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