Introduction to Probability
Probability is a measure of the likelihood of an event occurring.
Types of Probability
| Type | Description |
|---|---|
| Experimental (Empirical) | Based on actual experiments; P(E) = number of times event occurred / total trials |
| Theoretical | Based on equally likely outcomes; P(E) = favourable outcomes / total outcomes |
Class 10 focuses on theoretical probability.
Key Definitions
Trial: A single performance of an experiment (e.g., tossing a coin once).
Sample Space (S): The set of all possible outcomes of an experiment.
- Coin: S = {H, T}; n(S) = 2
- Die: S = {1, 2, 3, 4, 5, 6}; n(S) = 6
- Two coins: S = {HH, HT, TH, TT}; n(S) = 4
Event (E): A subset of the sample space.
Favourable outcomes: Outcomes that belong to the event E.
Probability Formula
$$P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Total number of equally likely outcomes}}$$
Key properties:
- 0 ≤ P(E) ≤ 1
- P(impossible event) = 0 (e.g., getting 7 on a die)
- P(certain/sure event) = 1 (e.g., getting a number ≤ 6 on a die)
- P(E) + P(not E) = 1 → P(E̅) = 1 − P(E)
Standard Experiments
Tossing a Coin
S = {H, T}; n(S) = 2
| Event | Favourable | Probability |
|---|---|---|
| Getting Head | {H} | 1/2 |
| Getting Tail | {T} | 1/2 |
Tossing Two Coins
S = {HH, HT, TH, TT}; n(S) = 4
| Event | Favourable | P |
|---|---|---|
| Two heads | {HH} | 1/4 |
| At least one head | {HH, HT, TH} | 3/4 |
| No head | {TT} | 1/4 |
| Exactly one head | {HT, TH} | 1/2 |
Rolling a Die
S = {1, 2, 3, 4, 5, 6}; n(S) = 6
| Event | Favourable | P |
|---|---|---|
| Getting 4 | {4} | 1/6 |
| Prime number | {2, 3, 5} | 1/2 |
| Even number | {2, 4, 6} | 1/2 |
| Number > 4 | {5, 6} | 1/3 |
| Multiple of 3 | {3, 6} | 1/3 |
Rolling Two Dice
n(S) = 36 (6 × 6 = 36 equally likely outcomes)
| Event | Count | P |
|---|---|---|
| Sum = 7 | {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} = 6 | 6/36 = 1/6 |
| Sum = 2 (doublet 1,1) | 1 | 1/36 |
| Doublets | {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)} = 6 | 1/6 |
| Sum ≥ 10 | {(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)} = 6 | 1/6 |
Playing Cards Problems
A standard deck has 52 cards:
| Suit | Colour | Cards |
|---|---|---|
| Hearts ♥ | Red | A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K |
| Diamonds ♦ | Red | A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K |
| Spades ♠ | Black | A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K |
| Clubs ♣ | Black | A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K |
Key counts:
- Red cards = 26; Black cards = 26
- Face cards (J, Q, K) = 12; Aces = 4
- Non-face cards = 40; Number cards (2–10) = 36
- King of hearts = 1 specific card
Sample problems:
Q. One card drawn at random from 52 cards. Find probability it is:
(a) A king: P = 4/52 = 1/13
(b) A red card: P = 26/52 = 1/2
(c) A face card: P = 12/52 = 3/13
(d) The queen of spades: P = 1/52
(e) Not an ace: P = (52−4)/52 = 48/52 = 12/13
(f) A black face card: (J, Q, K of spades and clubs) = 6; P = 6/52 = 3/26
Balls in a Bag Problems
Q. A bag contains 5 red, 4 blue, and 3 green balls. One ball drawn at random. Find:
(a) P(red) = 5/12 (b) P(not green) = (5+4)/12 = 9/12 = 3/4 (c) P(blue or green) = (4+3)/12 = 7/12
Numbers and Digits Problems
Q. A number is chosen at random from 1 to 50. Find P(multiple of 3 or 5).
Multiples of 3 up to 50: 3,6,9,...,48 → 16 numbers Multiples of 5 up to 50: 5,10,...,50 → 10 numbers Multiples of both (15): 15,30,45 → 3 numbers
By inclusion-exclusion: 16 + 10 − 3 = 23
P = 23/50
Board Exam Tips
- 4–5 marks from this chapter — shorter than others but reliable marks
- Cards problem appears almost every year — memorise: 52 total, 26 red, 26 black, 4 suits, 13 each, 12 face cards, 4 aces
- Always reduce fractions to lowest terms
- "At least one" type: use complement → P(at least one) = 1 − P(none)
- Two dice: n(S) = 36 always; list outcomes systematically to count favourable
- Common mistake: forgetting to include both (H,T) and (T,H) when order matters in two-coin problems
📚 RD Sharma Mathematics Class 10 — Probability Chapter
📝 Oswaal CBSE Maths Question Bank — Probability Board Questions