Probability
1. Basic Concepts
PROBABILITY is the measure of how LIKELY an event is to occur.
Key Terms
| Term | Definition | Example |
|---|---|---|
| Experiment | An action with uncertain outcomes | Tossing a coin |
| Outcome | A possible result of an experiment | Head or Tail |
| Event | A set of one or more outcomes | Getting a Head |
| Sample Space | The SET of ALL possible outcomes | {H, T} |
| Favourable Outcomes | Outcomes that satisfy the event | {H} for 'getting a Head' |
Probability Scale
| Probability | Meaning |
|---|---|
| 0 | Impossible event |
| Between 0 and 1 | Possible (more likely as it approaches 1) |
| 0.5 | Equally likely |
| 1 | Certain event |
2. Formula for Probability
P(Event) = Number of favourable outcomes / Total number of possible outcomes
Where: 0 ≤ P(Event) ≤ 1.
Rules
- P(Event) + P(Not Event) = 1
- P(Not Event) = 1 - P(Event)
3. Coin Experiments
Tossing a Single Coin
- Sample space: {H, T}. Total outcomes = 2.
- P(Head) = 1/2. P(Tail) = 1/2.
Tossing Two Coins Together
- Sample space: {HH, HT, TH, TT}. Total outcomes = 4.
Worked Example (ICSE 2024, 2 marks): 'Two coins are tossed together. Find the probability of getting (i) two heads, (ii) at least one head.'
Solution: Total outcomes = 4. (i) Favourable outcomes for two heads: {HH}. P = 1/4. (ii) Favourable outcomes for at least one head: {HH, HT, TH}. P = 3/4.
Key Points for Coins
- Each toss is INDEPENDENT.
- Order matters when counting outcomes: HT and TH are DIFFERENT outcomes.
4. Die Experiments
Throwing a Single Die
- A standard die has faces numbered 1, 2, 3, 4, 5, 6.
- Sample space: {1, 2, 3, 4, 5, 6}. Total outcomes = 6.
Worked Example (ICSE 2023, 2 marks): 'A die is rolled. Find the probability of getting (i) an even number, (ii) a number greater than 4.'
Solution: Total outcomes = 6. (i) Favourable (even): {2, 4, 6}. P = 3/6 = 1/2. (ii) Favourable (>4): {5, 6}. P = 2/6 = 1/3.
Throwing Two Dice
- Total outcomes = 6 × 6 = 36.
- Sample space: {(1,1), (1,2), ..., (6,6)}.
Example: P(sum = 7) = ? Favourable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Total = 6. P = 6/36 = 1/6.
5. Playing Cards (Basic)
A standard deck has 52 cards divided into:
- 4 suits: Hearts ♥ (13), Diamonds ♦ (13), Clubs ♣ (13), Spades ♠ (13).
- Red suits: Hearts and Diamonds (26 cards).
- Black suits: Clubs and Spades (26 cards).
- Face cards: King, Queen, Jack (12 cards total — 3 per suit).
- Ace: 4 cards (one per suit).
Basic Probabilities with Cards
- P(Heart) = 13/52 = 1/4.
- P(Face card) = 12/52 = 3/13.
- P(Red card) = 26/52 = 1/2.
- P(Ace) = 4/52 = 1/13.
6. Simple Events from Daily Life
Drawing a Ball from a Bag
'A bag contains 3 red balls, 5 blue balls, and 2 green balls. Find the probability of drawing a blue ball.' Total balls = 3 + 5 + 2 = 10. P(Blue) = 5/10 = 1/2.
Selecting a Student
'In a class of 40 students, 18 are girls. What is the probability that a randomly chosen student is a boy?' Boys = 40 - 18 = 22. P(Boy) = 22/40 = 11/20.
7. Equally Likely Outcomes
OUTCOMES are EQUALLY LIKELY if each outcome has the SAME chance of occurring.
- Coins: Head and Tail are equally likely (fair coin).
- Die: All 6 faces are equally likely (fair die).
- Cards: All 52 cards are equally likely (well-shuffled deck).
Not Equally Likely
- 'Probability of rain tomorrow' — NOT equally likely with no rain.
- 'Winning a lottery' — NOT equally likely with losing (many more losing tickets).
8. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Coin toss problems (1 or 2 coins) | 2-3 marks | Very High |
| Die throw problems | 2-3 marks | Very High |
| Card probability (simple) | 2 marks | Medium |
| Ball from bag | 2 marks | High |
| P(not event) = 1 - P(event) | 1-2 marks | Medium |
Common Mistakes
- Writing probability as a RATIO (3:4) instead of a FRACTION (3/4).
- Not reducing fractions to simplest form (e.g., writing 2/4 instead of 1/2).
- Forgetting to include BOTH (H,T) and (T,H) when tossing two coins.
- Saying probability > 1 (impossible — max is 1 for certain events).
Self-Test (5 Questions)
Q1. A coin is tossed. What is P(Tail)? (1 mark)
- A) 0
- B) 1/2
- C) 1
- D) 1/4
Q2. A die is rolled. Find P(odd number). (2 marks)
Q3. Two coins are tossed. Find P(exactly one head). (2 marks)
Q4. A bag has 4 red, 6 blue, 5 green marbles. P(not blue) = ? (2 marks)
Q5. A die is rolled. Find P(number divisible by 3). (2 marks)
Answers
A1. B) 1/2. A2. 3/6 = 1/2. (Odd numbers: 1, 3, 5.) A3. 2/4 = 1/2. (Exactly one head: HT, TH.) A4. Total = 15. Not blue = red + green = 9. P = 9/15 = 3/5. A5. 2/6 = 1/3. (Numbers divisible by 3: 3, 6.)
