Probability Class 11: Events, Axiomatic Probability and Solved Questions

What Probability is about

Weightage: This chapter is part of Unit V, Statistics and Probability, which the syllabus gives 12 of the 80 theory marks in the school annual exam (CBSE adds that there is no fixed chapter-wise weightage).Source: CBSE Mathematics (041) syllabus 2026-27, Class XI course structure, Unit V Statistics and Probability

Key concepts

Sample space and events

NCERT section 14.1 and 14.1.1

An experiment whose outcome cannot be predicted in advance, but whose possible outcomes are known, has a sample space S: the set of all outcomes. Examples: one die, S = {1, 2, 3, 4, 5, 6}; two coins, S = {HH, HT, TH, TT}.

An event is any subset E of S. If the outcome of a trial is in E, we say E has occurred.

• Throw a die and get 4: the event 'even number' = {2, 4, 6} has occurred; 'number greater than 4' = {5, 6} has not.

(The NCERT section on random experiments, outcomes and sample spaces is formative-only for 2026-27, but listing sample spaces remains the starting point of every event question.)

Worked example

Two coins are tossed. Write the sample space and the events A: 'at least one head' and B: 'exactly one tail'. [2 marks]

Try it first, then show the solutionSolution

1. S = {HH, HT, TH, TT}
2. A = {HH, HT, TH}; B = {HT, TH}

Answer: As listed; note B ⊂ A.

Types of events

NCERT section 14.1.2

• Impossible event: φ, which never occurs (e.g. 'a die shows 7').
• Sure event: S itself, which always occurs (e.g. 'a die shows a number less than 7').
• Simple (elementary) event: a single outcome, e.g. {5}. A sample space with n outcomes has n simple events.
• Compound event: more than one outcome, e.g. {2, 4, 6}.

With n outcomes, the number of possible events is $2^n$ (all subsets of S, including φ and S).

Worked example

A die is thrown. Classify: (a) 'a multiple of 7' (b) 'a number less than 7' (c) 'the number 3' (d) 'an odd number'. [2 marks]

Try it first, then show the solutionSolution

1. (a) φ: impossible; (b) S: sure
2. (c) {3}: simple; (d) {1, 3, 5}: compound

Answer: Impossible, sure, simple, compound.

Algebra of events: not, and, or

NCERT section 14.1.3

Events are sets, so set operations give new events:

• Not A (complement): $A^{\prime }=S-A$, outcomes not in A.
• A or B: $A\cup B$, at least one of them occurs.
• A and B: $A\cap B$, both occur.
• A but not B: $A-B=A\cap B^{\prime }$.

De Morgan's laws carry over: 'neither A nor B' is $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$, and 'not both' is $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$.

Worked example

A die is thrown. E = 'even number', F = 'number greater than 3'. Find E or F, E and F, not E, and E but not F. [2 marks]

Try it first, then show the solutionSolution

1. E = {2, 4, 6}, F = {4, 5, 6}: E ∪ F = {2, 4, 5, 6}; E ∩ F = {4, 6}
2. E′ = {1, 3, 5}; E − F = {2}

Answer: {2, 4, 5, 6}; {4, 6}; {1, 3, 5}; {2}

Mutually exclusive and exhaustive events

NCERT section 14.1.4 and 14.1.5

• Mutually exclusive: A and B cannot happen together: $A\cap B=\varphi$. E.g. 'odd' and 'even' on one die. Simple events are always mutually exclusive.
• Exhaustive: $E_1,E_2,\dots ,E_n$ together cover the sample space: $E_1\cup E_2\cup \cdots \cup E_n=S$. At least one must occur.
• Mutually exclusive and exhaustive: they split S into non-overlapping pieces (a partition): exactly one of them occurs.

An event and its complement are always mutually exclusive and exhaustive.

Worked example

A die is thrown. A = {1, 2}, B = {3, 4, 5}, C = {5, 6}, D = {6}. Are A, B, C mutually exclusive? Are they exhaustive? Do A, B, D form a partition? [2 marks]

Try it first, then show the solutionSolution

1. B ∩ C = {5} ≠ φ, so A, B, C are not mutually exclusive; A ∪ B ∪ C = S, so they are exhaustive
2. A, B, D are pairwise disjoint and their union is S: yes, a partition

Answer: Not mutually exclusive but exhaustive; A, B, D are mutually exclusive and exhaustive.

Axiomatic approach to probability

NCERT section 14.2 and 14.2.1

A probability on a sample space S is a function P that assigns a real number P(E) to each event E such that:
1. $P(E)\ge 0$ for every event E
2. $P(S)=1$
3. If E and F are mutually exclusive, $P(E\cup F)=P(E)+P(F)$

Consequences:

• $P(\varphi )=0$ (since $S=S\cup \varphi$ with disjoint parts).
• $0\le P(E)\le 1$ for every event.
• For outcomes ${\omega }_1,\dots ,{\omega }_n$: each $P({\omega }_i)\ge 0$ and $\sum P({\omega }_i)=1$; then $P(E)$ = sum of the probabilities of the outcomes in E.
• If $A\subset B$, then $P(A)\le P(B)$.

Valid assignment check: no negative value, none above 1, and the total exactly 1.

Worked example

For S = {a, b, c, d}, which assignment is valid? (i) 0.1, 0.2, 0.3, 0.4 (ii) 0.5, 0.5, 0.2, −0.2 (iii) 0.3, 0.3, 0.3, 0.3. [2 marks]

Try it first, then show the solutionSolution

1. (i) all ≥ 0, sum 1: valid. (ii) contains −0.2: invalid (although the sum is 1)
2. (iii) sum 1.2 ≠ 1: invalid

Answer: Only (i).

Equally likely outcomes (classical probability)

NCERT section 14.2.2

If S has n outcomes, all equally likely, then each has probability $\frac{1}{n}$ (by the axioms, they are equal and add to 1). For an event E with m outcomes:
$P(E)=\frac{n(E)}{n(S)}=\frac{m}{n}$

This is the classical definition from earlier classes, now a special case of the axiomatic one.

Counting help: use the counting principle and combinations from Chapter 6 for large sample spaces: e.g. the number of ways to choose 5 numbers out of 25 is ${}^{25}{C}_5=53130$.

Standard sample spaces: one die 6, two dice 36, n coins $2^n$, a pack of cards 52 (4 suits of 13; 26 red, 26 black; 12 face cards).

Worked example

Two dice are thrown. Find the probability that (a) the sum is 8 (b) the product is 6. [2 marks]

Try it first, then show the solutionSolution

1. n(S) = 36. (a) Sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2): $\frac{5}{36}$
2. (b) Product 6: (1, 6), (6, 1), (2, 3), (3, 2): $\frac{4}{36}=\frac{1}{9}$

Answer: $\frac{5}{36}$; $\frac{1}{9}$

Probability of 'or', 'and' and 'not' events

NCERT section 14.2.3 and 14.2.4

Addition rule (any two events):
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
We subtract $P(A\cap B)$ because outcomes in both A and B were counted twice.

• If A and B are mutually exclusive: $P(A\cup B)=P(A)+P(B)$.
• 'Not' rule: $P(A^{\prime })=1-P(A)$ (A and A′ are mutually exclusive and exhaustive).
• 'A but not B': $P(A-B)=P(A)-P(A\cap B)$.
• 'Neither': $P(A^{\prime }\cap B^{\prime })=1-P(A\cup B)$.
• Exactly one of A, B: $P(A)+P(B)-2P(A\cap B)$.

Tip: a Venn diagram with the four regions (A only, both, B only, neither) makes these problems quick.

Worked example

For two events, P(A) = 0.45, P(B) = 0.35 and P(A ∩ B) = 0.15. Find P(not A), P(A or B), P(A but not B) and P(neither). [3 marks]

Try it first, then show the solutionSolution

1. P(not A) = 1 − 0.45 = 0.55
2. P(A ∪ B) = 0.45 + 0.35 − 0.15 = 0.65; P(A − B) = 0.45 − 0.15 = 0.30
3. P(neither) = 1 − 0.65 = 0.35

Answer: 0.55, 0.65, 0.30, 0.35

Maths in real life: Probability

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Addition Rule, Complement and Bounds on Probability
• Classical Probability: Dice, Coins, Cards and Selections

Probability formula sheet

Axioms

$P(E)\ge 0,P(S)=1,P(E\cup F)=P(E)+P(F)\text{ if }E\cap F=\varphi$

Equally likely outcomes

$P(E)=\frac{n(E)}{n(S)}$

Addition rule

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

Complement

$P(A^{\prime })=1-P(A),P(\varphi )=0$

A but not B; neither

$P(A\cap B^{\prime })=P(A)-P(A\cap B),P(A^{\prime }\cap B^{\prime })=1-P(A\cup B)$

Exactly one of A and B

$P(A)+P(B)-2P(A\cap B)$

How the school exam asks Probability

The question patterns to expect, and the method that works for each.

Tricks to remember Probability

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Or = union, and = intersection, not = complement

  Translate words to set symbols before computing.

  Use it when: Algebra of events.

• Add, then remove the double count

  $P(A\cup B)=P(A)+P(B)-P(A\cap B)$: the overlap was counted twice.

  Use it when: Addition rule.

• Exclusive = no overlap; exhaustive = no gaps

  Mutually exclusive: nothing shared. Exhaustive: nothing left out.

  Use it when: Classifying events.

• Hard? Try the opposite

  'At least one' is often easiest as 1 − P(none).

  Use it when: Complement rule.

• Three checks for a valid P

  No negatives, none above 1, total exactly 1.

  Use it when: Validity questions.

Probability: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Probability

The slips examiners see most often, and what to write instead.

• ✗Mistake: Adding probabilities of overlapping events

  Forgetting the intersection.

  Wrong working

  P(king or heart) = $\frac{4}{52}+\frac{13}{52}=\frac{17}{52}$

  Right working

  $\frac{4+13-1}{52}=\frac{16}{52}=\frac{4}{13}$ (the king of hearts is in both)

  ✓Do this: Subtract P(A ∩ B) unless the events are mutually exclusive.

  Usually costs: 1-2 marks.

• ✗Mistake: Counting two dice outcomes as 21 instead of 36

  Treating (2, 5) and (5, 2) as the same.

  ✓Do this: The dice are different, so ordered pairs: 36 equally likely outcomes.

  Usually costs: Whole question.

• ✗Mistake: Calling a probability valid because it sums to 1

  Ignoring negative entries.

  ✓Do this: Check every value is between 0 and 1 as well.

  Usually costs: 1 mark.

• ✗Mistake: Confusing mutually exclusive with exhaustive

  Similar-sounding words.

  ✓Do this: Exclusive: $A\cap B=\varphi$. Exhaustive: union is S.

  Usually costs: 1 mark.

• ✗Mistake: Answering with a probability greater than 1

  Arithmetic or counting error.

  ✓Do this: Always sanity-check: $0\le P\le 1$.

  Usually costs: Whole answer.

• ✗Mistake: Wrong letter counts in 'letters of a word' questions

  Counting distinct letters instead of all letters.

  ✓Do this: n(S) is the total number of letters, with repeats; count vowels with repeats too.

  Usually costs: 1 mark.

NCERT exercise guide: Probability

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Probability

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Two dice are thrown. The probability of a doublet is:
   (a) $\frac{1}{36}$
   (b) $\frac{1}{6}$
   (c) $\frac{1}{12}$
   (d) $\frac{5}{36}$

   Hint

   6 doublets.

   Answer

   (b)

2. Q2 · 1 mark · MCQExam level

   If A and B are mutually exclusive with $P(A)=\frac{3}{5}$ and $P(B)=\frac{1}{5}$, then P(A or B) is:
   (a) $\frac{3}{25}$
   (b) $\frac{2}{5}$
   (c) $\frac{4}{5}$
   (d) 1

   Hint

   No overlap.

   Answer

   (c)

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): For a die, the events 'odd number' and 'even number' are mutually exclusive and exhaustive.
   Reason (R): They have no common outcome and together make up the sample space.
   (a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

   Hint

   Definitions.

   Answer

   (a)

4. Q4 · 1 mark · Very short answerExam level

   A coin is tossed three times. Find the probability of at least one tail.

   Hint

   Complement of HHH.

   Answer

   $1-\frac{1}{8}=\frac{7}{8}$.

5. Q5 · 2 marks · Very short answerHarder exam level

   A die is thrown. Find the probability of (i) a number at most 4 (ii) a multiple of 3 (iii) a number less than 7.

   Hint

   List each event.

   Answer

   (i) $\frac{4}{6}=\frac{2}{3}$ (ii) {3, 6}: $\frac{1}{3}$ (iii) sure event: 1.

6. Q6 · 2 marks · Very short answerHarder exam level

   Given P(A) = 0.35, P(B) = 0.45 and P(A ∪ B) = 0.6, find P(A ∩ B) and P(A′ ∩ B′).

   Hint

   Rearrange the addition rule.

   Answer

   P(A ∩ B) = 0.35 + 0.45 − 0.6 = 0.2; P(neither) = 1 − 0.6 = 0.4.

7. Q7 · 2 marks · Very short answerHarder exam level

   Four-digit numbers are formed from 1, 2, 3, 4, 5 without repetition. One is chosen at random. Find the probability that it is even.

   Hint

   Last digit 2 or 4.

   Answer

   Total $5\times 4\times 3\times 2=120$; even: $2\times 4\times 3\times 2=48$; $P=\frac{48}{120}=\frac{2}{5}$.

8. Q8 · 3 marks · Short answerHarder exam level

   From a group of 6 boys and 4 girls, 3 are chosen at random. Find the probability that exactly 2 of them are girls.

   Hint

   ${}^4{C}_2\times {}^6{C}_1$ favourable selections.

   Answer

   $\frac{{}^4{C}_2\times {}^6{C}_1}{{}^{10}{C}_3}=\frac{6\times 6}{120}=\frac{3}{10}$.

9. Q9 · 3 marks · Short answerHarder exam level

   In a school raffle, 4 different numbers are drawn from 1 to 15. What is the probability that a ticket with 4 chosen numbers matches all of them (order does not matter)?

   Hint

   ${}^{15}{C}_4$.

   Answer

   ${}^{15}{C}_4=\frac{15\times 14\times 13\times 12}{24}=1365$, so $P=\frac{1}{1365}$.

10. Q10 · 2 marks · Very short answerHOTS / competency

    Show that if $A\subset B$, then $P(A)\le P(B)$.

    Hint

    Split B.

    Answer

    B is the disjoint union of A and B − A, so P(B) = P(A) + P(B − A) ≥ P(A) since P(B − A) ≥ 0.

11. Q11 · 3 marks · Short answerHOTS / competency

    Three events A, B, C of a sample space are mutually exclusive and exhaustive, with P(B) = 2P(A) and P(C) = 3P(A). Find P(A), P(B) and P(C).

    Hint

    They add to 1.

    Answer

    $P(A)+2P(A)+3P(A)=1\Rightarrow P(A)=\frac{1}{6}$, $P(B)=\frac{1}{3}$, $P(C)=\frac{1}{2}$.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is an event?Show answer

   A subset of the sample space.

2. 2.How many events does a sample space with 3 outcomes have?Show answer

   $2^3=8$.

3. 3.When are A and B mutually exclusive?Show answer

   When $A\cap B=\varphi$.

4. 4.State the three axioms.Show answer

   P(E) ≥ 0; P(S) = 1; P(E ∪ F) = P(E) + P(F) for mutually exclusive E, F.

5. 5.What is P(φ)?Show answer

   0.

6. 6.State the addition rule.Show answer

   $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

7. 7.If P(A) = $\frac{3}{13}$, what is P(not A)?Show answer

   $\frac{10}{13}$.

8. 8.Probability of a doublet with two dice?Show answer

   $\frac{1}{6}$.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you write the sample space for two dice or three coins?
• ☐Can you list an event and its complement?
• ☐Can you classify events as impossible, sure, simple or compound?
• ☐Can you find A ∪ B, A ∩ B, A − B for given events?
• ☐Can you test whether events are mutually exclusive and exhaustive?
• ☐Can you state the axioms and prove P(φ) = 0?
• ☐Can you decide whether a probability assignment is valid?
• ☐Can you find probabilities with cards, dice and letters?
• ☐Can you use the addition rule and the complement rule together?
• ☐Can you find P(neither) and P(exactly one)?

Probability: frequently asked questions