Probability Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 14 notes: theoretical probability, the range of probability, complementary events, and problems on coins, dice and cards.
Probability — Class 10 Maths Notes
Chapter Snapshot
Probability measures how likely an event is. This chapter is about theoretical (classical) probability — favourable outcomes divided by total outcomes — its range from 0 to 1, complementary events, and the standard problem types on coins, dice, cards, and bags of balls.
Board relevance: an easy-to-score Probability chapter. Expect one or two direct problems. The key skill is counting outcomes correctly.
Key Concepts & Definitions
Random experiment — an action whose result is not known in advance (tossing a coin, rolling a die).
Outcome — a possible result. Sample space — the set of all possible outcomes. Event — a collection of outcomes.
Equally likely outcomes — outcomes with the same chance of happening (a fair coin, an unbiased die).
Elementary event — an event with a single outcome. The sum of the probabilities of all elementary events of an experiment is 1.
Formulas
Theoretical probability
P(E) = (number of outcomes favourable to E) / (total number of possible outcomes)
(assuming all outcomes are equally likely.)
Properties:
- 0 ≤ P(E) ≤ 1 — probability is never negative or more than 1.
- P(impossible event) = 0 (e.g. getting a 7 on a die).
- P(sure/certain event) = 1 (e.g. getting a number ≤ 6 on a die).
- Complementary events: P(not E) = 1 − P(E) — E and "not E" (written Ē) together cover all outcomes.
Standard total outcomes
Experiment Total outcomes
One coin 2 (H, T)
Two coins 4 (HH, HT, TH, TT)
One die 6 (1–6)
Two dice 36
Deck of cards 52
Deck of 52 cards: 4 suits (♥ hearts, ♦ diamonds — red; ♣ clubs, ♠ spades — black), 13 cards each; 26 red, 26 black; 12 face cards (K, Q, J of each suit); 4 aces.
Worked Examples
Example 1 — Die: A fair die is rolled. Find P(getting an even number).
Favourable = {2, 4, 6} = 3; total = 6. P = 3/6 = 1/2.
Example 2 — Cards: A card is drawn from a well-shuffled deck. Find P(a king) and P(a red card).
P(king) = 4/52 = 1/13. P(red) = 26/52 = 1/2.
Example 3 — Bag of balls: A bag has 5 red, 3 blue, and 2 green balls. Find P(not red).
Total = 10, red = 5, so not-red = 5. P(not red) = 5/10 = 1/2. (Check: P(red) = 1/2, and 1 − 1/2 = 1/2 ✓.)
Example 4 — Two coins: Two coins are tossed together. Find P(at least one head).
Sample space {HH, HT, TH, TT}; at least one head = {HH, HT, TH} = 3. P = 3/4. (Or 1 − P(no head) = 1 − 1/4 = 3/4.)
Example 5 — Two dice: Two dice are rolled. Find P(sum = 8).
Favourable: (2,6),(3,5),(4,4),(5,3),(6,2) = 5; total = 36. P = 5/36.
Example 6 — Cards after removal: All the face cards are removed from a deck. A card is drawn from the remaining 40. Find P(an ace).
Removing 12 face cards leaves 52 − 12 = 40 cards, including all 4 aces. P(ace) = 4/40 = 1/10. (Always recompute the total after cards are removed — a common trap.)
Example 7 — Find the number of balls: A bag has 12 balls, some white and the rest black. If P(white) = 1/3, find the number of black balls.
Let white = w. P(white) = w/12 = 1/3 → w = 4. So black = 12 − 4 = 8 balls.
Example 8 — Defective items: A box has 90 good and 10 defective bulbs. One bulb is drawn at random. Find P(defective) and P(good).
Total = 100. P(defective) = 10/100 = 1/10; P(good) = 1 − 1/10 = 9/10 (using the complement).
Important Question Patterns
1. Single die/coin (2 marks): P(even/odd/prime/greater than k); at least one head with two coins.
2. Cards (2–3 marks): P(king/queen/ace/red/black/face card/number card); after removing some cards, recompute the total.
3. Bag of balls (2–3 marks): P(a colour), P(not a colour) via the complement; find the number of balls given a probability.
4. Two dice (3 marks): P(a given sum), P(doublet), P(product/difference condition) — count from the 36 outcomes.
5. Complement/reasoning (1–2 marks): use P(not E) = 1 − P(E); why a probability can't be 1.5; sum of all elementary probabilities = 1.
⚡ Quick Revision
- P(E) = favourable outcomes / total outcomes (equally likely).
- 0 ≤ P(E) ≤ 1; impossible = 0, sure = 1.
- Complement: P(not E) = 1 − P(E) — use it when "at least one" or "not" appears.
- Sum of probabilities of all elementary events = 1.
- Totals: 1 coin → 2, 2 coins → 4, 1 die → 6, 2 dice → 36, deck → 52.
- Deck: 26 red + 26 black; 4 of each rank; 12 face cards; 4 aces.
- The whole skill is counting the favourable and total outcomes correctly.
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