Probability — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Probability (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Probability (CBSE & HBSE)
Probability is the final chapter and is scoring. CBSE tests classical probability with cards, dice, and real-life contexts. HBSE focuses on basic probability rules and standard experiments.
Basic Probability Concepts
Terminology
| Term | Definition | Example |
|---|---|---|
| Experiment | An action with uncertain outcomes | Tossing a coin |
| Sample Space (S) | Set of ALL possible outcomes | {H, T} for coin |
| Event (E) | A subset of sample space | Getting Head = {H} |
| Favourable outcomes | Outcomes that satisfy the event | 1 (for Head) |
| Equally likely outcomes | Each outcome has same probability | Fair coin/die |
| Certain event | Always occurs, P = 1 | Getting < 7 with a die |
| Impossible event | Never occurs, P = 0 | Getting 7 with a die |
Classical Probability
$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$
Important Facts:
- 0 ≤ P(E) ≤ 1 (probability always between 0 and 1)
- P(E) + P(Ē) = 1 where Ē = complement of E
- P(Ē) = 1 - P(E) (probability of NOT E)
Standard Sample Spaces
| Experiment | Sample Space | Size |
|---|---|---|
| Tossing 1 coin | {H, T} | 2 |
| Tossing 2 coins | {HH, HT, TH, TT} | 4 |
| Tossing 3 coins | {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} | 8 |
| Rolling 1 die | {1,2,3,4,5,6} | 6 |
| Rolling 2 dice | {(1,1),(1,2),...(6,6)} | 36 |
| Standard deck of cards | 52 cards | 52 |
Key Formula: P(not E) = 1 - P(E). If probability of event = 3/7, probability of NOT event = 4/7.
Playing Cards and Dice Problems
Standard Deck of 52 Cards
| Category | Details | Count |
|---|---|---|
| Suits | Hearts (♥), Diamonds (♦), Clubs (♣), Spades (♠) | 4 suits |
| Colour | Red (Hearts+Diamonds), Black (Clubs+Spades) | 26 each |
| Cards per suit | A,2,3,4,5,6,7,8,9,10,J,Q,K | 13 each |
| Face cards | J, Q, K (in each suit) | 12 total |
| Ace | 4 aces (one per suit) | 4 |
| Number cards | 2 through 10 | 9×4=36 |
| Honours/Picture | A,J,Q,K | 16 total |
Probability with Cards
- P(red card) = 26/52 = 1/2
- P(ace) = 4/52 = 1/13
- P(face card) = 12/52 = 3/13
- P(king of hearts) = 1/52
- P(red ace) = 2/52 = 1/26
Two Dice Sample Space (36 outcomes)
Sum problems:
| Sum | Favourable outcomes | Probability |
|---|---|---|
| 2 | (1,1) → 1 | 1/36 |
| 3 | (1,2),(2,1) → 2 | 2/36 = 1/18 |
| 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 | 6/36 = 1/6 |
| 8 | (2,6),(3,5),(4,4),(5,3),(6,2) → 5 | 5/36 |
| 12 | (6,6) → 1 | 1/36 |
Most probable sum = 7 (6 ways out of 36)
CBSE Exam Pattern: Card-based questions are 2-3 marks. Know all card categories by heart. Dice sum tables save time.
Real-Life Probability and Complementary Events
Real-Life Probability Problems
Many CBSE/HBSE problems use real-life scenarios:
- Selecting from a group of people
- Choosing defective/non-defective items
- Events at lottery draws or spinning wheels
Approach:
- Identify the sample space size (total outcomes)
- Identify favourable outcomes for the event
- Apply P(E) = favourable/total
Complementary Events
P(Ē) = 1 - P(E)
Use this when finding P(at least one) or P(not E):
- P(at least one head in 3 coin tosses) = 1 - P(no head) = 1 - P(TTT) = 1 - 1/8 = 7/8
Geometric Probability
When outcomes are from a geometric figure: $$P(\text{event}) = \frac{\text{Area/Length of favourable region}}{\text{Total Area/Length}}$$
Example: A point is randomly selected inside a circle with an inscribed square. Find P(point in square).
- P = Area of square / Area of circle
Probability on a Number Line
"A number is chosen from 1 to 50. Find P(multiple of 5)."
- Multiples of 5 from 1-50: {5,10,15,...,50} = 10 numbers
- P = 10/50 = 1/5
Frequency-Based (Empirical) Probability
$$P(E) = \frac{\text{Frequency of event}}{\text{Total observations}}$$ Used when actual data (frequency) given instead of theoretical model.
Theoretical vs Empirical: CBSE Class 10 uses classical (theoretical) probability based on equally likely outcomes.
Frequently asked questions
Are these Probability notes free?
Yes — the Probability notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Probability notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Probability chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Probability.