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Punjab (PSEB)Class 10 Mathematics★ Unit VII — Statistics and Probability: 11 marks, shared with Statistics · CBSE Class 10 Maths (Chapter 14)

Probability

Probability — the classical (theoretical) definition, sample space and events, and complementary events, applied to coins, dice, cards and everyday random draws. CBSE Class 10 Maths Chapter 14.

⏱ 20 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1State the classical (theoretical) definition of probability and apply it to a sample space
2Build the sample space correctly for coins, dice, cards and simple draws — the step most errors come from
3Use complementary events, P(E) + P(not E) = 1, especially for 'at least one' and 'not' questions
4Handle 'without replacement' problems where the sample space shrinks after the first draw
5Judge whether a proposed set of outcomes is actually equally likely, and say why or why not

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Why this chapter matters

This is the last chapter of Class 10 Maths, and one of the most directly useful: every 'what are the odds' question you'll ever ask — a card draw, a dice roll, a defective part in a batch — reduces to counting favourable outcomes over total outcomes.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Statistics

What CBSE examines here (2026-27). The classical definition of probability — favourable outcomes over total outcomes, assuming every outcome is equally likely — applied to coins, dice, cards, marbles and similar situations, plus complementary events. The chapter has always been a single exercise, 14.1, and still is. One question in it (the die-drop-on-a-rectangle geometric probability question) is marked by the textbook itself as "not from the examination point of view" — it is included here for completeness, clearly flagged.

1. About the Chapter

Probability quantifies UNCERTAINTY. It's used in:

Insurance (calculating risks)
Weather forecasting
Medicine (treatment effectiveness)
Sports analytics
Computer science (machine learning)
Cricket strike rates

Class 10 Focus

Mostly THEORETICAL probability (assuming equally likely outcomes).

Closes Class 10 Math

This is the final chapter of Class 10. Brief, but important foundation.

2. Basic Definitions

Random Experiment

An experiment whose outcome cannot be predicted with certainty.

Tossing a coin
Rolling a die
Drawing a card from deck

Outcome

A possible result of an experiment.

Tossing coin: Head (H) or Tail (T)
Rolling die: 1, 2, 3, 4, 5, or 6

Sample Space (S)

The set of ALL possible outcomes.

Tossing 1 coin: S = {H, T}
Rolling 1 die: S = {1, 2, 3, 4, 5, 6}

Event

A subset of the sample space.

'Getting a head' = {H}
'Getting an even number' = {2, 4, 6}

Favourable Outcomes

Outcomes in which the desired event occurs.

3. Probability Formula

Classical Probability

P(E) = (Number of favourable outcomes) / (Total number of outcomes)

= n(E) / n(S)

Range

0 ≤ P(E) ≤ 1

P(E) = 0: IMPOSSIBLE event
P(E) = 1: SURE/CERTAIN event
0 < P(E) < 1: Possible

Sum of Probabilities

For all outcomes in sample space: SUM = 1.

For an event E and its complement E': P(E) + P(E') = 1

So: P(E') = 1 − P(E)

4. Common Examples

Coin Toss

1 coin: S = {H, T}; P(H) = 1/2
2 coins: S = {HH, HT, TH, TT}; P(at least one H) = 3/4

Dice

1 die: S = {1, ..., 6}
P(prime) = 3/6 = 1/2 (primes: 2, 3, 5)
P(even) = 3/6 = 1/2
P(> 4) = 2/6 = 1/3

Two Dice

Total outcomes: 6 × 6 = 36
Possible sums: 2 to 12
P(sum = 7) = 6/36 = 1/6 (highest probability!)

Cards (52 in a deck)

4 suits: Hearts, Diamonds (red); Clubs, Spades (black)
13 cards per suit: A, 2, 3, ..., 10, J, Q, K
Face cards: J, Q, K (12 total: 3 × 4)
P(any specific card) = 1/52
P(ace) = 4/52 = 1/13
P(face card) = 12/52 = 3/13
P(red card) = 26/52 = 1/2
P(spade) = 13/52 = 1/4

5. Complementary Events

Definition

For event E, the complementary event E' (or NOT E) contains all outcomes NOT in E.

Formula

P(E) + P(NOT E) = 1

Example

If P(passing exam) = 0.8, then P(failing) = 1 − 0.8 = 0.2.

Useful When

Direct counting is hard
'At least one' problems

6. Worked Examples

Example 1: Single Die

A die is rolled. Find P(getting a number < 4).

Favourable: 1, 2, 3 (three outcomes)
Total: 6
P = 3/6 = 1/2

Example 2: Card Draw

A card is drawn from 52-deck. Find: (a) P(red card) = 26/52 = 1/2 (b) P(king) = 4/52 = 1/13 (c) P(red king) = 2/52 = 1/26 (d) P(card with letter on it) = 16/52 = 4/13 (Ace + J + Q + K)

Example 3: Two Coins

Two coins are tossed. Find P(exactly one head).

Sample space: {HH, HT, TH, TT}
Favourable: {HT, TH}
P = 2/4 = 1/2

Example 4: Box of Balls

A box contains 3 red, 5 blue, 2 green balls. A ball is drawn at random. (a) P(red) = 3/10 (b) P(blue) = 5/10 = 1/2 (c) P(not red) = 7/10 (d) P(red or blue) = 8/10 = 4/5

Example 5: Two Dice — Sum

Two dice rolled. Find P(sum = 8).

Total outcomes: 36
Favourable: (2,6), (3,5), (4,4), (5,3), (6,2) — 5 outcomes
P = 5/36

Example 6: Birthday Problem

A class has 30 students. Probability that 2 have same birthday?

Very HIGH (≈ 70%) — famous birthday paradox
(Beyond Class 10 syllabus but interesting)

7. Geometric Probability (Brief)

Marked non-examinable by the textbook itself. NCERT's Exercise 14.1 has one question on this idea (a die dropped at random on a rectangle, asking for the probability it lands inside a marked circle) — and the question is starred with a footnote reading "Not from the examination point of view." It is included here because it is genuinely instructive, not because it will be asked.

When outcomes lie in a geometric region: P = (Favourable region area) / (Total region area)

Example

A point is chosen randomly inside a square of side 2. Probability it lies in inscribed circle?

Circle area = π(1)² = π
Square area = 4
P = π/4 ≈ 0.785

8. Common Mistakes

Forgetting all outcomes equally likely
- Classical probability ASSUMES equal likelihood. For weighted dice, biased coins, this fails.
Wrong sample space
- For 2 coins, S = {HH, HT, TH, TT} (4 outcomes), not 3 (HH, HT, TT).
Adding probabilities of intersecting events
- P(A or B) ≠ P(A) + P(B) if events overlap. (Need inclusion-exclusion, Class 11.)
Confusing P(E) and P(E')
- P(E) + P(E') = 1, so P(NOT E) = 1 − P(E).
Probability > 1
- IMPOSSIBLE. If you get P > 1, you've made an error.

9. Real-World Applications

Insurance

Companies calculate probability of accidents, deaths to set premiums.

Weather

'70% chance of rain' = probability statement.

Cricket

Run rate, strike rate use probability concepts
Probability of winning given current score

Medicine

Probability of recovering from a disease
Drug effectiveness in trials

Gambling/Games

Lottery: P(winning) ≈ 1/14,000,000 (extremely low!)
Poker probabilities

Indian Context

IPL match predictions use probability
Election polling
Indian Statistical Institute leads probability research

10. Famous Probability Problems

Coin Toss Paradox

If you toss a fair coin 10 times and get 10 heads in a row, P(11th toss being head) = ?

Still 1/2!
Each toss is INDEPENDENT.

Monty Hall Problem

3 doors, prize behind one. You pick door 1. Host opens door 3 (empty). Should you switch to door 2?

YES! Switching increases probability from 1/3 to 2/3.
Famous and counterintuitive.

Birthday Paradox

In a room of 23 people, probability that 2 share a birthday is over 50%.

Counterintuitive but true.

11. Conclusion

Probability is the mathematics of UNCERTAINTY:

We can quantify how likely things are
Make informed decisions under uncertainty
Foundation for statistics, machine learning, gambling theory

Master:

Formula: P = favourable / total
Sample space for various experiments
Complementary events
Word problems (cards, dice, coins, real-life)

This chapter is BRIEF but IMPORTANT for board exam (~6-8 marks).

In Class 11-12, you'll learn:

Conditional probability
Bayes' theorem
Probability distributions
Statistics with probability

For now, master the basics. Practice 15+ problems.

Probability: the math of guessing right.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Classical probability

P(E) = (number of favourable outcomes) / (total number of equally likely outcomes)

The word 'equally likely' is doing real work — it is the assumption behind every formula in this chapter

Range of probability

0 ≤ P(E) ≤ 1

P(E) = 0 is an impossible event; P(E) = 1 is a certain (sure) event

Complementary events

P(E) + P(not E) = 1, so P(not E) = 1 − P(E)

The single most useful shortcut in the chapter — use it whenever 'not' or 'at least one' appears

Sum over the sample space

the probabilities of all elementary events add to 1

A quick check: if your listed outcomes don't sum to 1, something was miscounted

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Building the wrong sample space

✓ Two coins give 4 equally likely outcomes {HH, HT, TH, TT}, not 3 {HH, HT, TT} — HT and TH are genuinely different outcomes even though both show 'one head, one tail'.

WATCH OUT

✗ Forgetting a shrinking sample space in 'without replacement' questions

✓ Once a card, marble or bulb is removed and not put back, both the count of that item and the total count change for the next draw. Recompute both before finding the second probability.

WATCH OUT

✗ Assuming events are equally likely just because there are only two of them

✓ 'The car starts or doesn't' has two outcomes, but they are not equally likely — one may be far more probable than the other. Equally likely has to be justified, not assumed from the count of outcomes.

WATCH OUT

✗ Missing the complementary-event shortcut

✓ Questions asking for 'not E' or 'at least one' are almost always faster via 1 − P(complement) than by directly counting every favourable outcome.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 14.1

Exercise 14.1

Classical probability and complementary events — coins, dice, cards, marbles, defective items, and one non-examinable geometric-probability question

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Probability?

3 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

3 questions~2 min worth ~8 marks in Punjab (PSEB) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•P(E) = favourable outcomes / total outcomes, assuming every outcome is equally likely
•0 ≤ P(E) ≤ 1 always; P(E) = 0 impossible, P(E) = 1 certain
•P(E) + P(not E) = 1 — the complementary-events shortcut
•Build the sample space carefully — HT and TH are different outcomes, as are the two individual dice in a two-dice throw
•'Without replacement' shrinks both the favourable count and the total for the next draw
•This chapter has a single exercise, 14.1, with 25 questions
•One question (the geometric-probability die-drop) is explicitly marked non-examinable by the textbook

Punjab (PSEB) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
MCQ	1	2-3	Definitions, simple P
Short	2-3	1-2	Card, dice, coin problems
Long	5	0-1	Multi-step probability

Prep strategy

Master sample space for coins (2,4,8), dice (6,36), cards (52)
Practice 20+ problems
Use complementary events when easier

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Insurance

Premiums based on probability of accidents, deaths. Indian LIC uses actuarial probability.

Weather forecasts

'70% chance of rain' is a probability statement. IMD uses probabilistic models.

Cricket analytics

Win probability, strike rate analysis use probability.

Indian elections

Polling uses sample probability to predict outcomes.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write out the sample space explicitly before counting anything, especially for two coins or two dice

For 'without replacement' problems, redraw the total and favourable counts fresh for each subsequent draw

Reach for 1 − P(complement) whenever 'not' or 'at least one' appears in the question

Sanity-check: your answer must be between 0 and 1 — if it isn't, a count somewhere is wrong

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

Bayes' theorem

STRETCH

Random variables

STRETCH

Probability distributions

STRETCH

Monte Carlo methods

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 10 BoardHigh

Maths OlympiadMedium

JEE FoundationHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Yes — Probability is one of the chapters rationalisation left alone. It has always had a single exercise, 14.1, with 25 questions, and it still does. If a source tells you otherwise for this chapter, it is wrong.

That is the textbook's own footnote on the geometric-probability question (a die dropped at random on a rectangle containing a marked circle). NCERT flags it as background enrichment rather than examinable content — it is the one place in this chapter where 'probability' means a ratio of areas rather than a ratio of counts.

Whenever the question uses the words 'not', 'at least one', or describes an event that would need many separate cases to count directly. Find the probability of the opposite, simpler event, and subtract from 1.

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