Probability ISC Class 11: Sample Spaces, Events and the Addition Theorem

What Probability is about

Weightage: Probability belongs to the unit Statistics & Probability, which CISCE weights at 8 of the 80 marks in the ISC Class XI Mathematics Paper I (Theory). Class XI is examined by the school, so your school sets the paper within this weightage.Source: CISCE ISC Mathematics syllabus, examination year 2028, Class XI unit weightage

Key concepts

Random experiments, outcomes and sample spaces

A random experiment is one that can be repeated under the same conditions, has more than one possible outcome, and whose outcome cannot be predicted in advance. Each possible result is an outcome (sample point); the set of all outcomes is the sample space S. Write S carefully as a set, keeping order where the experiment has stages (first coin, second coin). The number of outcomes n(S) can be found with the counting principle.

Worked example

Write the sample space for (i) tossing a coin twice (ii) rolling a die and then tossing a coin. How many outcomes does each have?

Try it first, then show the solutionSolution

1. (i) S = {HH, HT, TH, TT}; n(S) = 4
2. (ii) S = {1H, 1T, 2H, 2T, …, 6H, 6T}; n(S) = 6 × 2 = 12

Answer: 4 outcomes; 12 outcomes

Events and their occurrence

An event E is a subset of S; E occurs when the outcome of the experiment is an element of E. A simple (elementary) event has one outcome; a compound event has more than one. S itself is the sure (certain) event, and ∅ the impossible event. Describing an event in words and then listing its outcomes is the core skill.

Worked example

Two dice are rolled. List the event 'the sum is 8'.

Try it first, then show the solutionSolution

1. Outcomes (first, second) with sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
2. n(E) = 5 out of n(S) = 36

Answer: E = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}

'Not', 'and', 'or' events

Events combine exactly like sets. Not A: A′ = S − A (A does not occur). A and B: A ∩ B (both occur). A or B: A ∪ B (at least one occurs). A but not B: A − B = A ∩ B′. Neither A nor B: A′ ∩ B′ = (A ∪ B)′ (De Morgan's law). Exactly one of A, B: (A − B) ∪ (B − A).

Worked example

A die is rolled. A: an even number, B: a prime number. Find A ∩ B, A ∪ B, A′ and A − B.

Try it first, then show the solutionSolution

1. A = {2, 4, 6}, B = {2, 3, 5}
2. A ∩ B = {2}; A ∪ B = {2, 3, 4, 5, 6}
3. A′ = {1, 3, 5}; A − B = {4, 6}

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

Mutually exclusive and exhaustive events

Events A and B are mutually exclusive if they cannot occur together: A ∩ B = ∅. Events E₁, E₂, …, Eₙ are exhaustive if at least one of them must occur: E₁ ∪ E₂ ∪ … ∪ Eₙ = S. Events that are both mutually exclusive (in pairs) and exhaustive split S into separate pieces; their probabilities add to 1. An event and its complement are always mutually exclusive and exhaustive.

Worked example

A die is rolled. Are A = {1, 2}, B = {3, 4}, C = {5, 6} mutually exclusive and exhaustive? What about A and D = {2, 3}?

Try it first, then show the solutionSolution

1. A ∩ B = B ∩ C = A ∩ C = ∅, and A ∪ B ∪ C = S: yes to both
2. A ∩ D = {2} ≠ ∅: A and D are not mutually exclusive

Answer: Yes; A and D are not mutually exclusive

Axiomatic definition of probability

A probability on a sample space S assigns to each event E a number P(E) such that: (1) P(E) ≥ 0; (2) P(S) = 1; (3) if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B). For a finite S, it is enough to give each outcome ωᵢ a weight P(ωᵢ) ≥ 0 with total 1; then P(E) is the sum of the weights of the outcomes in E. If all n(S) outcomes are equally likely, each weight is 1/n(S), and we recover the classical rule P(E) = n(E)/n(S) from earlier classes. Consequences: 0 ≤ P(E) ≤ 1 and P(∅) = 0.

Worked example

Find the probability that the sum is 8 when two fair dice are rolled.

Try it first, then show the solutionSolution

1. 36 equally likely outcomes; event has 5 outcomes
2. P = 5/36

Answer: 5/36 ≈ 0.139

Probability of 'not' an event

A and A′ are mutually exclusive and A ∪ A′ = S, so P(A) + P(A′) = P(S) = 1, giving P(not A) = 1 − P(A). This is the easiest route for 'at least one' questions: P(at least one) = 1 − P(none). Also, P(A − B) = P(A) − P(A ∩ B).

Worked example

A fair coin is tossed 4 times. Find the probability of at least one head.

Try it first, then show the solutionSolution

1. n(S) = 2⁴ = 16; 'no head' = {TTTT}, probability 1/16
2. P(at least one head) = 1 − 1/16

Answer: 15/16

Addition theorem

For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B): adding P(A) and P(B) counts the outcomes in A ∩ B twice, so we subtract once. If A and B are mutually exclusive, P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B). For three events, P(A ∪ B ∪ C) is the sum of the three single probabilities, minus the three pairwise intersection probabilities, plus P(A ∩ B ∩ C). Useful derived results: P(exactly one of A, B) = P(A) + P(B) − 2P(A ∩ B); P(neither) = 1 − P(A ∪ B).

Worked example

A card is drawn from a well-shuffled pack of 52. Find the probability that it is a king or a heart.

Try it first, then show the solutionSolution

1. P(king) = 4/52, P(heart) = 13/52, P(king of hearts) = 1/52
2. P = 4/52 + 13/52 − 1/52 = 16/52

Answer: 4/13

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 Theorem and Probability of Combined Events
• Sample Space, Events and Probability of an Event

Probability formula sheet

Classical probability

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

Only for equally likely outcomes.

Axioms

$P(E)\ge 0;P(S)=1;P(A\cup B)=P(A)+P(B)\text{ if }A\cap B=\varnothing$

All other rules follow from these.

Complement

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

P(at least one) = 1 − P(none).

Addition theorem

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

Mutually exclusive: last term is 0.

Three events

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

Add singles, subtract pairs, add the triple.

Only A; exactly one

$P(A\cap B^{\prime })=P(A)-P(A\cap B);P(\text{exactly one})=P(A)+P(B)-2P(A\cap B)$

Neither: 1 − P(A ∪ B).

Probability: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

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 adds, overlap subtracts

  P(A or B) = P(A) + P(B) − P(A and B).

  Use it when: Addition theorem.

• Not is one minus

  P(not A) = 1 − P(A).

  Use it when: Complements and 'at least one'.

• Exclusive means empty overlap

  Mutually exclusive ⇔ A ∩ B = ∅ ⇔ P(A ∩ B) = 0.

  Use it when: Spotting when to just add.

• Exhaustive means covers S

  Union of exhaustive events is the whole sample space.

  Use it when: Exhaustive events.

• Singles, pairs, triple

  Three events: + singles − pairs + triple.

  Use it when: Three-event addition.

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 D style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 5-mark question split into 1 + 1 + 1 + 2 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: Forgetting to subtract the overlap

  Students add P(A) and P(B) for events that can happen together.

  Wrong working

  P(king or heart) = 4/52 + 13/52 = 17/52.

  Right working

  Subtract the king of hearts: 16/52.

  ✓Do this: Ask 'can both happen at once?' before adding.

  Usually costs: 1-2 marks.

• ✗Mistake: Unordered outcomes treated as equally likely

  Students write two coins as {two heads, one head, no head}.

  Wrong working

  P(two heads) = 1/3.

  Right working

  S = {HH, HT, TH, TT}: P(two heads) = 1/4.

  ✓Do this: Keep track of which coin or die is which.

  Usually costs: Full marks for the part.

• ✗Mistake: A probability bigger than 1

  Arithmetic or double counting.

  Wrong working

  P(A ∪ B) = 0.6 + 0.7 = 1.3.

  Right working

  Use P(A ∩ B); and any answer must lie between 0 and 1.

  ✓Do this: Check every answer is in [0, 1].

  Usually costs: 1-2 marks.

• ✗Mistake: Confusing 'mutually exclusive' with 'exhaustive'

  The words sound similar.

  Wrong working

  {1, 2} and {5, 6} on a die are exhaustive because they do not overlap.

  Right working

  They are mutually exclusive but not exhaustive (3 and 4 are missing).

  ✓Do this: Exclusive: no overlap. Exhaustive: nothing left out.

  Usually costs: 1 mark.

• ✗Mistake: Counting 'at least one' directly and overcounting

  Students fix one defective item and choose the rest freely.

  Wrong working

  At least one defective in 2 from 20 (4 defective): 4 × 19 favourable.

  Right working

  1 − ¹⁶C₂/²⁰C₂ = 7/19 (76 counts a pair of defectives twice).

  ✓Do this: Use the complement.

  Usually costs: 2 marks.

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

   The probability of an impossible event is:
   (a) 0
   (b) 1
   (c) 1/2
   (d) −1

   Hint

   P(∅).

   Answer

   (a)

2. Q2 · 1 mark · MCQExam level

   If P(E) = 0.35, then P(not E) is:
   (a) 0.35
   (b) 0.65
   (c) 1.35
   (d) 0.5

   Hint

   1 − P(E).

   Answer

   (b)

3. Q3 · 1 mark · Very short answerExam level

   A coin is tossed and a die is rolled. How many outcomes are in the sample space?

   Hint

   2 × 6.

   Answer

   12

4. Q4 · 1 mark · Very short answerExam level

   A card is drawn from a pack of 52. Find P(red card).

   Hint

   26 red cards.

   Answer

   1/2

5. Q5 · 2 marks · Short answerExam level

   For a family with three children, write the sample space (B = boy, G = girl, in birth order) and find P(exactly two girls), assuming all outcomes are equally likely.

   Hint

   8 outcomes.

   Answer

   S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}; exactly two girls: BGG, GBG, GGB; P = 3/8

6. Q6 · 2 marks · Short answerHarder exam level

   A and B are mutually exclusive with P(A) = 2/7 and P(B) = 3/7. Find P(A ∪ B) and P(A′ ∩ B′).

   Hint

   Just add.

   Answer

   5/7; 2/7

7. Q7 · 2 marks · Short answerHarder exam level

   Two dice are rolled. Find the probability that the product of the numbers is 12.

   Hint

   List the pairs.

   Answer

   (2, 6), (3, 4), (4, 3), (6, 2): 4/36 = 1/9

8. Q8 · 2 marks · Short answerHarder exam level

   A letter is chosen at random from the word MATHEMATICS. Find the probability that it is (i) a vowel (ii) the letter M.

   Hint

   11 letters; vowels A, E, A, I.

   Answer

   (i) 4/11 (ii) 2/11

9. Q9 · 3 marks · Short answerHarder exam level

   Two cards are drawn at random from a pack of 52. Find the probability that both are aces.

   Hint

   Use combinations.

   Answer

   ⁴C₂/⁵²C₂ = 6/1326 = 1/221

10. Q10 · 2 marks · Short answerHarder exam level

    P(A ∪ B) = 0.7, P(A) = 0.4 and P(B) = 0.5. Find P(A ∩ B) and P(A only).

    Hint

    Rearrange the addition theorem.

    Answer

    P(A ∩ B) = 0.4 + 0.5 − 0.7 = 0.2; P(A only) = 0.4 − 0.2 = 0.2

11. Q11 · 3 marks · Short answerHarder exam level

    A number is chosen at random from 1 to 30. Find the probability that it is a prime or a multiple of 4.

    Hint

    Can a prime be a multiple of 4?

    Answer

    Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (10); multiples of 4: 7; none in common, so P = 17/30

12. Q12 · 3 marks · Short answerHarder exam level

    Three people are chosen at random from 5 men and 4 women. Find the probability that exactly two are women.

    Hint

    ⁴C₂ × ⁵C₁.

    Answer

    30/84 = 5/14

13. Q13 · 5 marks · Long answerHarder exam level

    P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3.
    (a) Find P(A ∪ B). [1]
    (b) Find P(A′ ∩ B′). [1]
    (c) Find P(A′ ∪ B′). [1]
    (d) Find P(exactly one of A, B). [2]

    Hint

    De Morgan: A′ ∪ B′ = (A ∩ B)′.

    Answer

    (a) 0.8 (b) 0.2 (c) 1 − 0.3 = 0.7 (d) 0.6 + 0.5 − 2(0.3) = 0.5

14. Q14 · 3 marks · Short answerHOTS / competency

    In a survey of 100 people, 50 read paper A, 40 read B, 30 read C, 15 read A and B, 10 read B and C, 12 read A and C, and 5 read all three. Find the probability that a person chosen at random reads at least one paper, and none.

    Hint

    Three-event addition theorem.

    Answer

    (50 + 40 + 30 − 15 − 10 − 12 + 5)/100 = 88/100 = 0.88; none 0.12

15. Q15 · 3 marks · Short answerHOTS / competency

    Can P(A) = 0.6, P(B) = 0.7 and P(A ∩ B) = 0.2 for two events of the same experiment? Give a reason.

    Hint

    Find P(A ∪ B).

    Answer

    No: P(A ∪ B) = 0.6 + 0.7 − 0.2 = 1.1 > 1, which is impossible. (P(A ∩ B) must be at least 0.3.)

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 a sample space?Show answer

   The set of all possible outcomes of a random experiment.

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

   When A ∩ B = ∅.

3. 3.When are events exhaustive?Show answer

   When their union is S.

4. 4.Three axioms of probability?Show answer

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

5. 5.P(not A)?Show answer

   1 − P(A).

6. 6.Addition theorem?Show answer

   P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

7. 7.'Neither A nor B' as a set?Show answer

   A′ ∩ B′ = (A ∪ B)′.

8. 8.P(exactly one of A, B)?Show answer

   P(A) + P(B) − 2P(A ∩ B).

Quick revision checklist

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

• ☐Can you write the sample space for two- and three-stage experiments?
• ☐Can you list an event such as 'sum is 9' for two dice?
• ☐Can you express 'neither', 'exactly one' and 'A but not B' with sets?
• ☐Can you test whether events are mutually exclusive and exhaustive?
• ☐Can you state the axioms of probability?
• ☐Can you use the complement for 'at least one'?
• ☐Can you apply the addition theorem for two and three events?
• ☐Can you spot impossible combinations of given probabilities?

Probability: frequently asked questions