Probability — Class 10 Notes

CBSE 2027 — dates, rules and official papers →

Maths · Ch 14Class 10 · CBSE 2026–27Medium weightage

Probability

What the board actually asks: single-event probability from a bag, die, coin or a deck of cards, the complement rule P(not E) = 1 - P(E), and the impossible-0 / sure-1 / 0 to 1 range facts - together these carry almost every mark in this chapter.

NCERT VERBATIM sourced, word-for-wordAUTHORED LazyTopper teaching scaffolding

The Big Idea AUTHORED

Count the wanted, count the total, divide.

Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain). When every outcome of an experiment is equally likely, the theoretical (classical) probability of an event $E$ is just $P (E) = \frac{favourable outcomes}{all possible outcomes}$ . The whole chapter is that one idea applied to coins, dice, cards and bags of marbles - plus one shortcut, the complement $P (not E) = 1 - P (E)$ , and one sanity check, $0 \leq P (E) \leq 1$ .

Formulae

Classical probability

P (E) = \frac{Number of outcomes favourable to E}{Number of all possible outcomes}

Tip: List the sample space and count favourable outcomes.

Complement rule

P (E) = 1 - P (E)

Tip: "at least one" → 1 − P(none).

Range of probability

0 \leq P (E) \leq 1

Tip: answers outside 0–1 are wrong.

Impossible / sure event

P (impossible) = 0 \cdot P (sure) = 1

Elementary events

\sum P (elementary events) = 1

Tip: Sum of probabilities of all elementary events = 1.

Playing Cards: What You Need

52 cards in 4 suits of 13 each: spades and clubs are black, hearts and diamonds are red (26 red, 26 black). Each suit has ace, king, queen, jack and 2 to 10. Kings, queens and jacks are the face cards: 12 in all. There are 4 aces.

Coins and Dice: Sample Spaces

One coin: 2 outcomes. Two coins: 4. Three coins: 8. One die: 6. Two dice thrown together: 36.

Equally Likely Outcomes

The formula

P (E) = \frac{favourable outcomes}{all possible outcomes}

assumes that all outcomes are equally likely.

Shortcuts & memory tricks →

Definitions the board rewards NCERT VERBATIM

State these in NCERT’s exact words

Theoretical (classical) probability

NCERT verbatim — memorise this exact wording· Ch 14,

“The theoretical probability (also called classical probability) of an event E, written as P(E), is defined as P(E) = Number of outcomes favourable to E / Number of all possible outcomes of the experiment, where we assume that the outcomes of the experiment are equally likely.”

What it means:

P (E) = \frac{outcomes favourable to E}{all possible outcomes}

. It ONLY applies when the outcomes are equally likely - check that first, or the formula is invalid.

Elementary event

NCERT verbatim — memorise this exact wording· Ch 14,

“An event having only one outcome of the experiment is called an elementary event.”

What it means: an event with exactly ONE outcome (e.g. 'the die shows 5'). Across an experiment, the probabilities of all elementary events add up to

1

Complement / complementary events

NCERT verbatim — memorise this exact wording· Ch 14,

“The event E̅, representing 'not E', is called the complement of the event E. We also say that E and E̅ are complementary events.”

What it means: 'not

E

' is everything in the sample space outside

E

. Because

E

and

E

together fill the whole space,

P (E) = 1 - P (E)

- the workhorse shortcut for 'not' and 'at least one' questions.

Impossible event

NCERT verbatim — memorise this exact wording· Ch 14,

“the probability of an event which is impossible to occur is 0. Such an event is called an impossible event.”

What it means: an event with NO favourable outcome (e.g. getting an

8

on an ordinary die). Its probability is exactly

0

Sure (certain) event

NCERT verbatim — memorise this exact wording· Ch 14,

“the probability of an event which is sure (or certain) to occur is 1. Such an event is called a sure event or a certain event.”

What it means: an event whose favourable outcomes are ALL the outcomes (e.g. getting a number less than

7

on a die). Its probability is exactly

1

Key terms — NCERT verbatim (state word-for-word)

Equally likely outcomes: We refer to this by saying that the outcomes head and tail, are equally likely. (Ch 14, )
Sum of elementary events = 1: the sum of the probabilities of all the elementary events of an experiment is 1. (Ch 14, )
Range of P(E): the numerator (number of outcomes favourable to the event E) is always less than or equal to the denominator (the number of all possible outcomes). (Ch 14, )

Must-know concepts AUTHORED FRAMING

…and where they’re tested

Classical probability = favourable / total

List every outcome of the experiment, confirm they are equally likely, count the favourable ones, and divide:

P (E) = \frac{favourable}{total}

. Always simplify the fraction.

1-mark MCQ + 2-3 mark 'find the probability' (bag / die / coin / cards)

Sample space and elementary events

The sample space is the set of all possible outcomes. An elementary event has just one outcome. A useful check: the probabilities of all the elementary events of an experiment add to

1

listing outcomes for one/two coins or dice before counting

Complementary events - the shortcut

E

and 'not

E

' together cover the whole sample space, so

P (not E) = 1 - P (E)

. When a question says 'not', 'at least one', or gives you the opposite event, the complement is usually the fast route.

'not an ace', 'at least one head', 'not a 5' - complement saves counting

Impossible (0), sure (1), and the range

An impossible event has probability

0

; a sure event has probability

1

; every probability lies in between:

0 \leq P (E) \leq 1

. A value below

0

or above

1

can never be a probability.

'which of these cannot be a probability?' MCQ (e.g. -1.5, 15%, 0.7)

Two-dice / two-coin sample spaces

For two dice, list ordered pairs - order matters, so

(1, 4) \neq = (4, 1)

- giving

6 \times 6 = 36

equally likely outcomes. Then count the pairs favourable to the event. Two coins give

{(H, H), (H, T), (T, H), (T, T)}

FigureTwo dice (blue and grey): the 36 equally likely ordered pairs, with the five cells whose sum is 8 shaded.

Figure asset pending extraction — the sourced original is Figure.

Every cell is one equally likely outcome. P(sum = 8) = 5/36 - count the shaded anti-diagonal and divide by the 36 total cells.

'sum on two dice' (NCERT Example 13) - a Section C/D fixture

Worked examples — real solved problems NCERT VERBATIM

Solved exactly as the source does them

Every problem is a real sourced item, quoted verbatim with its page; the step-by-step decoding and mark-logic are LazyTopper-authored coaching.

Ch 14, Example 1,

“Find the probability of getting a head when a coin is tossed once. Also find the probability of getting a tail.”

Write the sample space.Tossing a coin once has two possible outcomes - Head (H) and Tail (T) - and they are equally likely.Concept: list equally likely outcomes

Probability of a head.Let

E

= 'getting a head'. Favourable outcomes

= 1

, total

= 2

, so

P (head) = \frac{1}{2}

.Concept: favourable / total

1 mark

Probability of a tail.Let

F

= 'getting a tail'. Favourable outcomes

= 1

, total

= 2

, so

P (tail) = \frac{1}{2}

.Concept: favourable / total

1 mark

Mark-scheme logic (authored): 2-mark shape: 1 for P(head) = 1/2 and 1 for P(tail) = 1/2, each with the sample space named. Writing the outcomes {H, T} is the method line and is not separately credited.

Ch 14, Example 3,

“Suppose we throw a die once. (i) What is the probability of getting a number greater than 4 ? (ii) What is the probability of getting a number less than or equal to 4 ?”

Ch 14, Example 4,

“One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will (i) be an ace, (ii) not be an ace.”

Ch 14, Example 5,

“Two players, Sangeeta and Reshma, play a tennis match. It is known that the probability of Sangeeta winning the match is 0.62. What is the probability of Reshma winning the match?”

Ch 14, Example 8,

“A box contains 3 blue, 2 white, and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will be (i) white? (ii) blue? (iii) red?”

Ch 14, Example 13,

“Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) 13? (iii) less than or equal to 12?”

Problem-solving strategies AUTHORED

'Find the probability'?Count favourable outcomes, count ALL equally likely outcomes, divide, then simplify the fraction.

'not' / 'at least one'?Use the complement:

P (not E) = 1 - P (E)

- usually far quicker than counting the event directly.

Two dice or two coins?Write the full sample space first -

36

ordered pairs for dice,

4

for two coins - then count favourable.

'Which cannot be a probability'?Reject anything below

0

or above

1

. A negative value or one greater than

1

is impossible.

A deck of cards?

52 = 4

suits

\times 13

26

red,

26

black,

12

face cards,

4

of each rank. Pick out the count you need.

Common mistakes that cost marks

Where students lose easy marks

Applying the formula when outcomes are not equally likely - e.g. 'the car starts or does not start'. The classical formula only holds for equally likely outcomes.

Treating two-dice outcomes as unordered -

(1, 4)

and

(4, 1)

are different, so the sample space has

36

pairs, not

21

Not writing 0 for an impossible event or 1 for a sure event - these are exact numbers, not 'no answer'.

Leaving the fraction unsimplified, or giving a probability above 1 - always reduce, and sanity-check that

0 \leq P (E) \leq 1

Formulae

Classical probability

P (E) = \frac{Number of outcomes favourable to E}{Number of all possible outcomes}

Tip: List the sample space and count favourable outcomes.

Complement rule

P (E) = 1 - P (E)

Tip: "at least one" → 1 − P(none).

Range of probability

0 \leq P (E) \leq 1

Tip: answers outside 0–1 are wrong.

Impossible / sure event

P (impossible) = 0 \cdot P (sure) = 1

Elementary events

\sum P (elementary events) = 1

Tip: Sum of probabilities of all elementary events = 1.

Playing Cards: What You Need

Coins and Dice: Sample Spaces

One coin: 2 outcomes. Two coins: 4. Three coins: 8. One die: 6. Two dice thrown together: 36.

Equally Likely Outcomes

The formula

P (E) = \frac{favourable outcomes}{all possible outcomes}

assumes that all outcomes are equally likely.

Shortcuts & Tricks

Any answer outside [0, 1] is wrong.

MCQ / check only: don't write this as working

"not E" → 1 − P(E); "at least one" → 1 − P(none).

Safe in written answers

Face cards = 12 (J, Q, K × 4), not 16.

MCQ / check only: don't write this as working

Write the sample space, count favourable and total.

Safe in written answers

Two dice: 36 outcomes; doubles 6.

MCQ / check only: don't write this as working

Memory Tricks

Anukool upon Kul.

P(E) = favourable / total. Anukool = favourable outcomes; Kul = all possible outcomes.

Zero = namumkin, ek = pakka.

P=0 impossible; P=1 sure.

4 suit (hukum, paan, eent, chidi) × 13 patte = 52; Raja, Rani, Gulam × 4 suit = 12 face cards.

4 suits (spades, hearts, diamonds, clubs; 2 black, 2 red) × 13 cards = 52; K, Q, J in each suit = 12 face cards.

Mindmap

- - P(E) = favourable / all outcomes
  - Assumes outcomes equally likely
  - Always simplify the fraction
- - List every equally likely outcome
  - Elementary event = one outcome
  - Sum of all elementary probs = 1
- - P(not E) = 1 - P(E)
  - 'not' / 'at least' → use complement
  - E and (not E) fill the whole space
- - Impossible event → 0
  - Sure event → 1
  - 0 <= P(E) <= 1 always
- - Coin(s): H, T
  - Die: 1-6
  - Cards: 52 = 4 x 13
  - Two dice: 36 ordered pairs
- - Outcomes must be equally likely
  - (1,4) != (4,1): 36 not 21
  - Give 0 / 1, never a blank
  - Never P > 1 or P < 0

Why 0 <= P(E) <= 1, and why P(E) + P(not E) = 1

These two facts are not memorised rules - they fall straight out of the definition P(E) = favourable / total. NCERT derives them by reasoning about the favourable count against the total count (pp.206-207). Understanding the argument means you never doubt a bound or a complement in the exam.

Start from the definition of probability.

P (E) = \frac{favourable outcomes}{all possible outcomes}

The classical definition, valid because the outcomes are equally likely (p.203).

The favourable count is never negative, and the total count is positive, so P(E) is never below 0.

P (E) \geq 0

The least it can be is 0, when NO outcome is favourable - an impossible event, e.g. getting an 8 on a die: 0/6 = 0 (pp.206-207).

The favourable outcomes are part of all the outcomes, so their count is at most the total; hence P(E) is never above 1.

P (E) \leq 1

NCERT: the numerator is always less than or equal to the denominator. The most it can be is 1, when EVERY outcome is favourable - a sure event, e.g. a number less than 7 on a die: 6/6 = 1 (p.207).

Combining the two bounds gives the range of any probability.

0 \leq P (E) \leq 1

Stated in NCERT immediately after the numerator/denominator argument (p.207).

Now the complement: 'not E' is exactly the outcomes NOT favourable to E, so the two favourable counts add up to the total.

(favourable to E) + (favourable to not E) = total

E and 'not E' together cover the whole sample space with no overlap (p.206).

Divide through by the total count to get the complement rule.

P (E) + P (E) = 1 \Rightarrow P (E) = 1 - P (E)

NCERT: P(E) + P(not E) = 1, which gives P(not E) = 1 - P(E) (p.206).

3 competency questions from the CBSE board pattern, with the full step-marked solution for each — the marks are shown per step, as the official scheme awards them.

2 marksSection B

Diwali Fest is an annual South Asian arts & culture festival produced by the Diwali Celebration Society. In the Diwali fest, a game is played with a fair spinner (shown). The numbers on the spinner are 2, 5, 7, 9, 12, 16. Sometimes the owner will invite a player who does not win with the spinner to throw the dice as a free bonus. What is the probability that a player will get a special prize because the spinner stops on a perfect square? Show your working.

Figure from the question paper for: Diwali Fest is an annual South Asian arts & culture festival produced by the Diwali Celebr

Step-marked solution

[1 mark] Identify perfect squares among 2, 5, 7, 9, 12, 16: these are 9 and 16 → 2 favourable out of 6.
[1 mark] P(perfect square) = 2/6 = 1/3 (or 0.33).

2 marksSection B

In the Diwali fest game, if the player gets a chance to throw a dice (as a free bonus after not winning with the fair spinner), what is the probability of getting a multiple of 2 on the dice? Show your working.

Step-marked solution

[1 mark] Identify outcomes that are multiples of 2 on a die: 2, 4 and 6 → 3 favourable out of 6.
[1 mark] P(multiple of 2) = 3/6 = 1/2 (or 0.5).

5 marksSection D

In a game the entry fee is ₹5. The game consists of tossing a coin 3 times. If one or two heads show, Sweta gets her entry fee back; if she throws 3 heads she gets double the entry fee; otherwise she loses. For tossing a coin three times, find the probability that she (i) loses the entry fee, (ii) gets double the entry fee, (iii) just gets her entry fee back.

Step-marked solution

[1 mark] The coin is tossed 3 times, so the sample space has 2³ = 8 equally likely outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
[1 mark] (i) She loses only when no head shows (0 heads) — outcome TTT, i.e. 1 favourable outcome.
[1 mark] (i) P(loses the entry fee) = 1/8.
[1 mark] (ii) She gets double the fee only on 3 heads (HHH) — 1 outcome, so P(gets double) = 1/8.
[1 mark] (iii) She gets her fee back on exactly 1 or 2 heads = 8 − 1(HHH) − 1(TTT) = 6 outcomes, so P = 6/8 = 3/4.

Practise Probability competency questions

Read NCERT Chapter 14 on LazyTopper (jemh114)

Official NCERT chapter PDF (jemh114) — ncert.nic.in

Sources: NCERT 2026-27 (tap to view)

Source ledger PROVENANCE

Source edition: NCERT Reprint 2026-27, Class 10 Mathematics, Ch 14 (jemh114.pdf, printed pages 202-217). One row per sourced item — every verbatim, example and NCERT figure traces to its page.

Item	Type	Source
Theoretical (classical) probability	verbatim	NCERT 2026-27 Ch14
Elementary event	verbatim	NCERT 2026-27 Ch14
Complement / complementary events	verbatim	NCERT 2026-27 Ch14
Impossible event	verbatim	NCERT 2026-27 Ch14
Sure (certain) event	verbatim	NCERT 2026-27 Ch14
Equally likely outcomes	verbatim	NCERT 2026-27 Ch14
Sum of elementary events = 1	verbatim	NCERT 2026-27 Ch14
Range of P(E)	verbatim	NCERT 2026-27 Ch14
Coin toss - a head, and a tail - NCERT Example 1	example	NCERT 2026-27 Ch14 Example 1
Single die - grouped events - NCERT Example 3	example	NCERT 2026-27 Ch14 Example 3
Cards - ace / not-ace via the complement - NCERT Example 4	example	NCERT 2026-27 Ch14 Example 4 -208
Complementary events - a tennis match - NCERT Example 5	example	NCERT 2026-27 Ch14 Example 5
Marbles - three colours - NCERT Example 8	example	NCERT 2026-27 Ch14 Example 8
Two dice - sum of the faces - NCERT Example 13	example	NCERT 2026-27 Ch14 Example 13 -213

Probability · spec-rendered note · NCERT Reprint 2026-27, Class 10 Mathematics, Ch 14 (jemh114.pdf, printed pages 202-217)

Practise writing answers the way examiners mark them → Guide