Chapter 14 of 14

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Rajasthan (RBSE)Class 10 Mathematics★ 2–4 marks · RBSE Class 10 Maths board (Unit — Statistics & Probability)

Probability

RBSE Class 10 Maths on the theoretical (classical) approach to probability — favourable outcomes over total outcomes — with dice, cards, coins and marbles. Learn the 0-to-1 rule, complementary events, and sure/impossible events, with fully solved board-pattern questions.

⏱ 24 min readEasy difficulty✓ Free · NCERT-aligned

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

1Apply the classical definition P(E) = favourable/total
2Use the rules 0 ≤ P ≤ 1, and P(sure)=1, P(impossible)=0
3Use complementary events P(not E) = 1 − P(E)
4List sample spaces for coins, dice and cards
5Solve word problems with marbles, cards, dice and coins

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Why this chapter matters

One of the easiest scoring chapters. With dice/coin/card sample spaces memorised, probability questions are quick and reliable — often 2–4 marks with minimal calculation and no proofs.

Probability — RBSE Class 10 (Mathematics)

Toss a coin, roll a die, draw a card — you cannot say what will happen, but you can say how likely it is. Probability puts a precise number, between 0 and 1, on chance. Class 10 focuses on the clean theoretical approach: count the favourable outcomes, count all the equally-likely outcomes, and divide.

1. The theoretical (classical) definition

When all outcomes of an experiment are equally likely, the probability of an event $E$ is $P (E) = \frac{number of outcomes favourable to E}{total number of possible outcomes}$

Everything in this chapter is careful counting of these two numbers.

2. The rules probability must obey

$0 \leq P (E) \leq 1$ for every event.
$P (sure event) = 1$ (it always happens); $P (impossible event) = 0$ .
The probabilities of all outcomes add to 1.
Complement: $P (not E) = 1 - P (E)$ . Written $P (\overset{ˉ}{E}) = 1 - P (E)$ — a huge time-saver for "at least" questions.

3. The standard sample spaces

Know these totals cold:

One coin: 2 outcomes {H, T}. Two coins: 4 outcomes {HH, HT, TH, TT}.
One die: 6 outcomes {1,…,6}. Two dice: 36 ordered outcomes.
A pack of 52 cards: 26 red + 26 black; 4 suits (♠♣ black, ♥♦ red) of 13; face cards = J, Q, K → 12 in all; aces = 4.

4. Worked micro-examples

A die is rolled. $P (even) = \frac{3}{6} = \frac{1}{2}$ ; $P (prime) = \frac{3}{6} = \frac{1}{2}$ (primes 2, 3, 5).
One card is drawn. $P (king) = \frac{4}{52} = \frac{1}{13}$ ; $P (red face card) = \frac{6}{52} = \frac{3}{26}$ .
Two dice are thrown. $P (sum = 7) = \frac{6}{36} = \frac{1}{6}$ (the pairs (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)).

5. Method that never fails

Write the total number of equally-likely outcomes.
Count the favourable outcomes carefully (list them for dice/coins if unsure).
Divide and simplify the fraction.
For "at least one / not" questions, use the complement $1 - P (\overset{ˉ}{E})$ .

A frequent word-problem type: "a marble is drawn from a bag of x red, y white, z blue…" — the total is $x + y + z$ , and "not red" uses the complement.

6. Closing thought

Probability in Class 10 is honest arithmetic wrapped around careful counting — no calculus, no guesswork. Get the sample-space totals for coins, dice and cards fixed in memory, reach for the complement on "at least/not" questions, and always simplify. The RBSE board reliably offers 2–4 easy marks here, so it is among the best return-on-effort topics in the paper.

Key formulas & results

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

Classical probability

P(E) = favourable outcomes / total outcomes

Outcomes must be equally likely.

Range

0 ≤ P(E) ≤ 1

Never negative or above 1.

Sure / impossible

P(sure)=1, P(impossible)=0

Extremes of the scale.

Complement

P(not E) = 1 − P(E)

Best for 'at least/not' questions.

Total sample spaces

coin 2, two coins 4, die 6, two dice 36, deck 52

Memorise these.

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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

✗ Wrong total number of outcomes

✓ Two dice give 36 (ordered) outcomes, two coins give 4. Fix the total before counting favourable cases.

WATCH OUT

✗ Miscounting face cards

✓ Face cards are J, Q, K only = 12 in a deck. The ace is NOT a face card.

WATCH OUT

✗ Giving probability greater than 1 or negative

✓ Every probability lies in [0, 1]. If you get 7/6, recount — the total is wrong.

WATCH OUT

✗ Not simplifying the fraction

✓ Reduce to lowest terms, e.g. 6/36 = 1/6, unless a decimal is asked.

WATCH OUT

✗ Doing 'at least one' the long way

✓ Use the complement: P(at least one) = 1 − P(none) — far quicker.

NCERT exercises (with solutions)

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

Ex 15.1

Exercise 15.1

Theoretical probability with coins, dice, cards and marbles

Questions

Examples (in-text)

Worked probability problems, including complementary events

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Die

A die is rolled. What is the probability of getting an even number?

▶ Show solution

Step 1 — Even numbers: 2, 4, 6 (3 favourable), total 6.

Step 2 — P = 3/6 = 1/2.

✦ Answer: 1/2.

Q2EASY· Card

One card is drawn from a well-shuffled deck. Find P(getting a king).

▶ Show solution

Step 1 — 4 kings out of 52 cards.

Step 2 — P = 4/52 = 1/13.

✦ Answer: 1/13.

Q3EASY· Complement

If P(E) = 0.35, find P(not E).

▶ Show solution

Step 1 — P(not E) = 1 − 0.35 = 0.65.

✦ Answer: 0.65.

Q4MEDIUM· Marbles

A bag has 5 red, 8 white and 7 blue marbles. One is drawn at random. Find P(not white).

▶ Show solution

Step 1 — Total = 20; white = 8, so not-white = 12.

Step 2 — P(not white) = 12/20 = 3/5.

✦ Answer: 3/5.

Q5MEDIUM· Two dice

Two dice are thrown. Find the probability that the sum is 7.

▶ Show solution

Step 1 — Total outcomes = 36. Favourable: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6.

Step 2 — P = 6/36 = 1/6.

✦ Answer: 1/6.

Q6MEDIUM· Card face

A card is drawn from a deck. Find P(a red face card).

▶ Show solution

Step 1 — Red face cards = J, Q, K of hearts and diamonds = 6.

Step 2 — P = 6/52 = 3/26.

✦ Answer: 3/26.

Q7MEDIUM· Two coins

Two coins are tossed together. Find the probability of getting at least one head.

▶ Show solution

Step 1 — Sample space {HH, HT, TH, TT}; P(no head) = P(TT) = 1/4.

Step 2 — P(at least one head) = 1 − 1/4 = 3/4.

✦ Answer: 3/4.

Q8HARD· Find number

A bag has 12 balls, some white. If the probability of drawing a white ball is 3 times that of a non-white ball, find the number of white balls.

▶ Show solution

Step 1 — Let white = w, so non-white = 12 − w.

Step 2 — w/12 = 3 × (12−w)/12 ⇒ w = 3(12 − w) = 36 − 3w.

Step 3 — 4w = 36 ⇒ w = 9.

✦ Answer: 9 white balls.

Q9HARD· Two dice product

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

▶ Show solution

Step 1 — Total = 36. Products = 12: (2,6),(6,2),(3,4),(4,3) = 4 outcomes.

Step 2 — P = 4/36 = 1/9.

✦ Answer: 1/9.

Q10HARD· Cards removed

All kings and queens are removed from a deck. A card is drawn from the remaining 44. Find P(getting a red card).

▶ Show solution

Step 1 — Removed 4 kings + 4 queens = 8, leaving 44. Red removed = 2 kings + 2 queens = 4, so red left = 26 − 4 = 22.

Step 2 — P = 22/44 = 1/2.

✦ Answer: 1/2.

5-minute revision

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

•P(E) = favourable/total for equally-likely outcomes.
•0 ≤ P(E) ≤ 1; sure event 1, impossible event 0.
•P(not E) = 1 − P(E).
•One die 6 outcomes, two dice 36; one coin 2, two coins 4.
•Deck: 52 cards, 26 red/26 black, 12 face cards (J,Q,K), 4 aces.
•Sum-of-two-dice: P(7) = 1/6 is the most common outcome.
•Always simplify the probability fraction.

Rajasthan (RBSE) marks blueprint

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

Typical chapter weightage: 2–4 marks

Question type	Marks each	Typical count	What it tests
MCQ / very short	1	1	Simple die/card probability; complement
Short answer	2	1	Marbles, two dice, two coins
Short answer	3	0–1	Find-the-number or modified-deck problem

Prep strategy

Memorise sample-space totals (coins, dice, cards)
Reach for the complement on 'at least/not' questions
List outcomes for two-dice sum/product problems to avoid miscounts
Always simplify the final fraction

Where this shows up in the real world

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

Games and lotteries

Odds in dice, cards and lotteries are direct probability calculations.

Insurance and risk

Premiums are based on the probability of claims occurring.

Weather forecasting

A '70% chance of rain' is a probability estimate.

Quality control

The chance a randomly picked item is defective guides sampling.

Exam strategy

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

State the total number of outcomes before counting favourable ones.
List outcomes explicitly for two-dice and two-coin problems.
Use the complement for 'not' and 'at least' questions.
Simplify every probability to lowest terms.
Check the answer lies between 0 and 1.

Going beyond the textbook

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

Addition and multiplication rules for combined events.
Conditional probability and independence.
Counting with permutations and combinations for larger sample spaces.
Expected value of a simple game.

Where else this chapter is tested

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

RBSE Class 10 Board (BSER Ajmer)High — 2–4 easy probability marks almost every year

NTSE / state scholarshipMedium — probability MCQs

SSC / competitive examsMedium — basic probability appears in quantitative sections

JEE FoundationMedium — foundation for Class 11–12 probability

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Questions students ask

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

Yes — RBSE (BSER, Ajmer) prescribes the NCERT Mathematics textbook; chapters match the national syllabus while RBSE sets its own exam pattern.

Theoretical (classical) probability is favourable/total assuming equally-likely outcomes. Experimental probability is based on actual trial results. Class 10 focuses on the theoretical approach.

No. Face cards are only the Jack, Queen and King — 12 in a deck. The four aces are counted separately.

Use the complement: P(at least one) = 1 − P(none). It is almost always faster than listing every favourable case.

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