Quick Summary
Probability measures the likelihood of an event occurring. Class 9 introduces experimental (empirical) and theoretical probability, sample spaces, types of events, and standard problems with coins, dice, and playing cards. This is a short but high-scoring chapter.
Key Definitions
- Random Experiment: An experiment whose outcome cannot be predicted in advance.
- Sample Space (S): Set of all possible outcomes of an experiment.
- Event (E): A subset of the sample space.
- Outcome: A single possible result of an experiment.
- Favourable Outcome: Outcomes that satisfy the event.
- Probability: P(E) = n(E) / n(S).
Types of Events
| Type | Definition | Example |
|---|---|---|
| Certain Event | Always occurs, P = 1 | Getting a number ≤ 6 on a die |
| Impossible Event | Never occurs, P = 0 | Getting 7 on a die |
| Complementary Events | E and E' together give S | Head and Tail on a coin |
| Mutually Exclusive | Cannot occur together | Getting 2 and 5 in one die roll |
| Equally Likely | Each has same probability | Any face of a fair die |
Standard Sample Spaces
One coin: S = {H, T} → n(S) = 2
Two coins: S = {HH, HT, TH, TT} → n(S) = 4
Three coins: S = 8 outcomes → n(S) = 8
One die: S = {1, 2, 3, 4, 5, 6} → n(S) = 6
Two dice: n(S) = 36
Standard deck of cards: 52 cards — 4 suits (♥ ♦ ♣ ♠), 13 cards each, 26 red + 26 black
Key Probability Rules
- 0 ≤ P(E) ≤ 1
- P(certain event) = 1
- P(impossible event) = 0
- P(E) + P(E') = 1
- P(E₁ or E₂) = P(E₁) + P(E₂) for mutually exclusive events
Important Questions
One Mark
1. A die is thrown once. What is P(getting an even number)? 2. Define a sample space. 3. What is the sum of probabilities of all outcomes in a sample space? 4. A bag has 3 red and 2 blue balls. What is P(drawing a red ball)? 5. Find P(E') if P(E) = 0.35.
Short Answer — 3 marks
1. Two coins are tossed simultaneously. Find P(at least one head), P(exactly two heads), P(no head). 2. A card is drawn from a well-shuffled deck of 52 cards. Find P(king), P(red card), P(face card). 3. A die is rolled twice. Find P(sum = 7) and P(sum = 12).
Long Answer — 5 marks
1. A bag contains 5 red, 3 blue, and 2 green balls. A ball is drawn at random. Find: (a) P(red), (b) P(blue), (c) P(not green), (d) P(red or blue), (e) Verify that the sum of all probabilities equals 1. 2. Two dice are thrown simultaneously. List the sample space and find P(same number on both dice), P(sum > 8), P(sum ≤ 3), P(one die shows 4).
Common Mistakes
- Not listing all outcomes in the sample space — always count n(S) first.
- Confusing P(E) = 0 (impossible) with P(E) = 1 (certain).
- Forgetting that P(not E) = 1 − P(E), not 1 + P(E).
- Double-counting in "at least one" problems — use complement: P(at least one) = 1 − P(none).
Revision Checklist
- [ ] Know the basic probability formula P(E) = n(E)/n(S)
- [ ] Can list sample spaces for coins and dice correctly
- [ ] Know the range: 0 ≤ P(E) ≤ 1
- [ ] Can find P(E') using P(E) + P(E') = 1
- [ ] Know standard facts about a 52-card deck
- [ ] Practised at least 3 problems with two dice (36 outcomes)