Class 10 · Chapter 14 · Statistics and Probability unit (11 of 80 marks)

Probability Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 53 board-style questions.

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53 questions
15 multiple choice, 5 assertion–reason, 12 very short answer, 12 short answer, 6 long answer, 3 case study
About 8 hours to master

Statistics & Probability unit: 11 of 80 theory marks (Statistics and Probability).

NCERT textbook: Chapter 14 on ncert.nic.in (PDF)

Revision notes

Theoretical (classical) probability

If an experiment has $n$ equally likely outcomes, and $m$ of them are favourable to event $E$ , then

$P (E) = \frac{number of outcomes favourable to E}{total number of possible outcomes} = \frac{m}{n}$ .

$0 \leq P (E) \leq 1$ . A sure event has probability 1 and an impossible event has probability 0.
Complementary events: $P (E) + P (\bar{E}) = 1$ , where $\bar{E}$ is "not $E$ ".
An event with a single outcome is an elementary event. The probabilities of all elementary events of an experiment add to 1.

Standard sample spaces

One coin: {H, T}. Two coins: {HH, HT, TH, TT}. Three coins: 8 outcomes.
One die: {1, 2, 3, 4, 5, 6}. Two dice: 36 ordered pairs $(a, b)$ . Sum 7 occurs 6 times, the most of any sum.
A pack of 52 cards: 4 suits (spades ♠ and clubs ♣ black; hearts ♥ and diamonds ♦ red), 13 cards each. Face cards are king, queen and jack: 12 in all. There are 4 aces.

Worked example 1

Two dice are thrown. Find the probability that the sum is 9. Favourable: (3,6), (4,5), (5,4), (6,3), so $P = \frac{4}{36} = \frac{1}{9}$ .

Worked example 2

A card is drawn from a well-shuffled pack. Find $P (a black face card)$ . There are 6 black face cards, so $P = \frac{6}{52} = \frac{3}{26}$ .

Worked example 3

A bag has 6 red balls and some blue balls. The probability of drawing a blue ball is $\frac{3}{5}$ . How many blue balls are there?

$\frac{b}{6 + b} = \frac{3}{5} \Rightarrow 5 b = 18 + 3 b \Rightarrow b = 9$ .

Common errors

Treating "HT" and "TH" as one outcome, or listing (2,3) but not (3,2) for two dice.
Forgetting that 1 is neither prime nor composite.
Counting 16 face cards (the aces are not face cards).
Not reducing the fraction, or giving a probability greater than 1.

Board-exam tips

Write the total number of outcomes and list the favourable outcomes explicitly. Each step earns a mark.
Use $P (not E) = 1 - P (E)$ when "not", "at least one" or "neither" is asked.
Only the classical (theoretical) approach is examined. The optional Exercise 14.2 has been removed from the rationalised book.

Topics in this chapter: Basic properties of probability · Probability with dice · Probability with coins · Probability with playing cards · Simple events · Complementary events.

Formulas for this chapter: Chapter 14 on the Class 10 formula sheet (free PDF and A3 poster).

Probability Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 145 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceCoins and dice

A fair die is rolled once. What is the probability that the number shown is a factor of $12$ ?

Show worked answer

Factors of $12$ on a die: $1, 2, 3, 4, 6$ : $P = \frac{5}{6}$ A1
Correct option: (c) $\frac{5}{6}$

Q2·1 mark·Multiple choicePlaying cards

A card is taken at random from a full pack of $52$ . What is the probability that it is a black card numbered from $2$ to $10$ ?

Q3·1 mark·Assertion–reasonCoins and dice

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): When two dice are thrown, the probability that the sum of the numbers is $13$ is $0$ .

Reason (R): The probability of an impossible event is $0$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerCoins and dice

A boy tosses two fair coins at once. What is the probability that he gets one head and one tail?

Show worked answer

Answer:

\frac{1}{2}

Equally likely outcomes $H H, H T, T H, T T$ ; one of each: $H T, T H$ M1
$P = \frac{2}{4} = \frac{1}{2}$ A1

Q5·2 marks·Short answerClassical probability

A bag holds $4$ red balls, $6$ blue balls and some green balls. A ball is drawn at random and the probability that it is green is $\frac{1}{3}$ . How many green balls are in the bag?

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerCoins and dice

Two fair dice are thrown at the same time. Find the probability that (a) the sum is $8$ , (b) the product is $12$ , (c) both dice show the same number.

Show worked answer

Answer:

\frac{5}{36}

\frac{1}{9}

\frac{1}{6}

$36$ equally likely outcomes. (a) $(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)$ : $\frac{5}{36}$ A1
(b) $(2, 6), (3, 4), (4, 3), (6, 2)$ : $\frac{4}{36} = \frac{1}{9}$ A1
(c) $6$ doubles: $\frac{6}{36} = \frac{1}{6}$ A1

Q7·3 marks·Short answerClassical probability

A box holds $50$ discs numbered $1, 2, 3, \dots, 50$ . One disc is taken out without looking. Find the probability that its number is (a) a multiple of $7$ , (b) a perfect square, (c) prime.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerPlaying cards

All the red face cards are removed from a pack of $52$ playing cards. A card is then drawn at random from the rest. Find the probability that it is (a) a king, (b) a red card, (c) a face card, (d) a spade, (e) the queen of hearts.

Show worked answer

Answer:

\frac{1}{23}

\frac{10}{23}

\frac{3}{23}

\frac{13}{46}

0

$6$ red face cards are removed, leaving $46$ cards M1
(a) $2$ kings left (both black): $\frac{2}{46} = \frac{1}{23}$ ; (b) $26 - 6 = 20$ red cards: $\frac{20}{46} = \frac{10}{23}$ A1
(c) $6$ black face cards: $\frac{6}{46} = \frac{3}{23}$ A1
(d) all $13$ spades remain: $\frac{13}{46}$ A1
(e) the queen of hearts was removed: probability $0$ A1

Q9·5 marks·Long answerCoins and dice

A fair die is thrown twice. Find the probability that (a) the first number is greater than the second, (b) the sum of the two numbers is a prime number, (c) the product of the two numbers is odd.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyClassical probability

Fun-fair token jar. At a school fun fair, a jar holds $20$ tokens: $8$ red, $7$ blue and $5$ yellow. A player draws one token at random.

Find the probability that the token is red. [1 mark]
Find the probability that the token is not yellow. [1 mark]
Two red tokens have already been drawn and not put back. Find the probability that the next token drawn is red. [2 marks]

Show worked answer

$P (red) = \frac{8}{20} = \frac{2}{5}$ A1
$\begin{aligned} P (not yellow) & = 1 - \frac{5}{20} \\ = \frac{3}{4} \end{aligned}$ A1
$18$ tokens are left, of which $6$ are red M1
$P (red) = \frac{6}{18} = \frac{1}{3}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

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 know that probabilities lie between 0 and 1, can use P(not E) = 1 − P(E), and can find simple probabilities.

Read first: Theoretical (classical) probability 12 practice questions · checkpoint: 4 questions, 7 marks, pass 80%

Practise step 1

Step 2

Board standard

You can list outcomes for coins, dice and cards correctly and solve the standard board questions on each.

Read first: Standard sample spaces; Worked examples 1-2 12 practice questions · checkpoint: 4 questions, 10 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write complete long answers and case studies, listing favourable outcomes and giving answers in lowest terms.

Read first: Standard sample spaces; Worked example 3; Board-exam tips 7 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle two-dice products and sums, modified card packs and 'find the number of balls' problems.

Read first: Standard sample spaces; Common errors 6 practice questions · checkpoint: 3 questions, 11 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. 6 of the 53 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceBasic properties of probability

A die is thrown once. Which of the following is an impossible event?

(a)getting 6
(b)getting 0
(c)getting an odd number
(d)getting a number less than 7

Q2·1 mark·Multiple choiceBasic properties of probability

The probability of a sure event is

(a)0
(b)1
(c) $\frac{1}{2}$
(d)more than 1

Q3·1 mark·Assertion–reasonBasic properties of probability

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The probability of getting a number greater than 6 in a single throw of a die is 0.

Reason (R): The probability of an impossible event is 0.

(a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c)Assertion (A) is true but Reason (R) is false.
(d)Assertion (A) is false but Reason (R) is true.

Worked solution·2 marks·Very short answerProbability with playing cards

Priya takes the six red face cards (the red jacks, queens and kings) out of a full pack of $52$ cards, shuffles the rest and draws one card. Find the probability that it is (a) a black king, (b) red.

Full worked solution Free, no sign-in

$6$ cards removed, $46$ left. (a) $\frac{2}{46} = \frac{1}{23}$ A1
(b) $\frac{26 - 6}{46} = \frac{10}{23}$ A1

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

Stretch yourself: original probability problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I138 · Problem I29 · Problem I126 · Problem I133. All 26 →

Themed lesson: Halloween maths lesson · Children’s Day (India) maths lesson · Pi Day maths lesson (starter, activity and worked answers). All themed maths

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

Where marks are lost in Probability

Treating HT and TH (or (2, 3) and (3, 2)) as one outcome. Fix: write the full sample space (4 for two coins, 36 for two dice) before counting.
Counting 16 face cards by including aces. Fix: face cards are J, Q, K only: 12 in a pack; aces are not face cards.
Forgetting that 1 is neither prime nor composite. Fix: list the primes explicitly when counting.
Not reducing the fraction, or giving a probability above 1. Fix: simplify every answer and check 0 ≤ P ≤ 1.
Not writing the total number of outcomes and the favourable list. Fix: write 'Total outcomes = 36, favourable = {(4, 5), (5, 4), …} = 4'; each part is marked.
Modified packs (cards removed): using 52 as the total. Fix: recount the total and each category after removal, on paper.

Exam strategy: how answers are step-marked · competency-based questions

Probability in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 17, 18, 25
Sample paper 2 (Board standard): questions 18, 38
Sample paper 3 (Above board): questions 18, 25, 31
Sample paper 4 (Above board): questions 18, 35, 38
Sample paper 5 (95+ challenge): questions 18, 25, 31

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).

Probability: common questions

What is in CBSE Class 10 Maths Chapter 14, Probability?

The topics are: Basic properties of probability, Probability with dice, Probability with coins, Probability with playing cards, Simple events, Complementary events. This page has revision notes for each, and 53 board-style practice questions with step-marked solutions.

How many marks is Probability worth?

It is in the Statistics and Probability unit, which carries 11 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Probability?

Treating HT and TH (or (2, 3) and (3, 2)) as one outcome. Fix: write the full sample space (4 for two coins, 36 for two dice) before counting. Counting 16 face cards by including aces. Fix: face cards are J, Q, K only: 12 in a pack; aces are not face cards.

Are the Probability important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

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.

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Revision notes

Theoretical (classical) probability

Standard sample spaces

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Probability Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Probability

Probability in our sample papers

Probability: common questions

Work through the Route to 95 with your progress saved