Probability
1. Basic Terms
| Term | Definition | Example |
|---|---|---|
| Experiment | An action with UNCERTAIN outcomes | Tossing a coin |
| Outcome | A POSSIBLE result of an experiment | Heads or Tails |
| Event | A set of OUTCOMES we are interested in | Getting Heads |
| Sample Space (S) | ALL possible outcomes | {H, T} |
| Trial | ONE performance of the experiment | One coin toss |
'Probability measures HOW LIKELY an event is to occur. It ranges from 0 (impossible) to 1 (certain).'
2. Theoretical Probability
P(Event) = Number of FAVOURABLE outcomes / Total number of POSSIBLE outcomes
P(E) = n(E) / n(S)
Formula: P(E) = Favourable outcomes / Total outcomes
Properties:
- 0 ≤ P(E) ≤ 1 (probability is always between 0 and 1)
- P(E) + P(not E) = 1
- If P(E) = 0, the event is IMPOSSIBLE
- If P(E) = 1, the event is CERTAIN
3. Experimental (Empirical) Probability
Based on ACTUAL experiments and observations.
P(E) = Number of times event OCCURRED / Total number of TRIALS
Law of Large Numbers: As the number of trials INCREASES, experimental probability APPROACHES theoretical probability.
4. Coin Problems
A coin has TWO outcomes: {Head, Tail}
| Event | Probability |
|---|---|
| Getting Heads | 1/2 |
| Getting Tails | 1/2 |
| Getting Heads or Tails | 1 (CERTAIN) |
| Getting NEITHER Heads nor Tails | 0 (IMPOSSIBLE) |
Worked Example: Two coins are tossed simultaneously. Find the probability of: (a) Two Heads (b) Exactly one Head (c) At least one Head
Sample Space S = {HH, HT, TH, TT}, n(S) = 4
(a) P(Two Heads) = {HH} = 1/4 (b) P(Exactly one Head) = {HT, TH} = 2/4 = 1/2 (c) P(At least one Head) = {HH, HT, TH} = 3/4
5. Dice Problems
A die has SIX outcomes: {1, 2, 3, 4, 5, 6}
| Event | Probability |
|---|---|
| Getting an EVEN number | 3/6 = 1/2 |
| Getting an ODD number | 3/6 = 1/2 |
| Getting a number > 4 | 2/6 = 1/3 |
| Getting a PRIME number | 3/6 = 1/2 (2, 3, 5 are prime) |
| Getting a number 1 | 1/6 |
| Getting a COMPOSITE number | 2/6 = 1/3 (4, 6 are composite) |
Worked Example: Two dice are thrown simultaneously. Find the probability of: (a) Sum = 7 (b) Sum > 9 (c) Doublet (same number on both)
n(S) = 6 × 6 = 36
(a) Sum = 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} = 6 outcomes P(Sum = 7) = 6/36 = 1/6
(b) Sum > 9: {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} = 6 outcomes P(Sum > 9) = 6/36 = 1/6
(c) Doublets: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} = 6 outcomes P(Doublet) = 6/36 = 1/6
6. Card Problems
A standard deck has 52 cards: 4 SUITS (Spades ♠, Hearts ♥, Diamonds ♦, Clubs ♣).
Each suit has 13 cards: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K.
| Event | Number of Favourable | Probability |
|---|---|---|
| Drawing a Heart | 13 | 13/52 = 1/4 |
| Drawing a King | 4 | 4/52 = 1/13 |
| Drawing a RED card | 26 | 26/52 = 1/2 |
| Drawing a FACE card (J, Q, K) | 12 | 12/52 = 3/13 |
| Drawing an ACE | 4 | 4/52 = 1/13 |
| Drawing a BLACK QUEEN | 2 | 2/52 = 1/26 |
Worked Example: A card is drawn from a well-shuffled deck. Find the probability of: (a) Getting a Spade (b) Not getting a Spade
(a) P(Spade) = 13/52 = 1/4 (b) P(Not Spade) = 1 — 1/4 = 3/4
7. Probability of 'Not' an Event
P(not E) = 1 — P(E)
Worked Example: If the probability of rain tomorrow is 0.35, what is the probability of NO rain?
P(No rain) = 1 — 0.35 = 0.65
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Probability can be greater than 1' | Probability is ALWAYS between 0 and 1 (inclusive) |
| 'Heads is more likely than Tails' | Both are EQUALLY likely — P(H) = P(T) = 1/2 |
| 'If I got 3 Heads in a row, Tails is more likely next' | Each toss is INDEPENDENT. P(Tails) is STILL 1/2. The coin has no memory |
| 'P(Even on a die) = number of evens / number of odds = 3/3 = 1' | Use TOTAL outcomes: P(Even) = 3/6 = 1/2 |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Probability of simple events (coin, die)
- 3-mark questions: One coin/die/card problem
- 4-mark questions: Two dice or two coins problems
- 6-mark questions: Card problems or combined events
Self-Test
Q1. A coin is tossed once. Find the probability of getting Heads. A1. P(Heads) = 1/2.
Q2. A die is thrown once. Find the probability of getting a number greater than 4. A2. Favourable: {5, 6}. P = 2/6 = 1/3.
Q3. Two coins are tossed. Find the probability of getting exactly one Tail. A3. Favourable: {HT, TH}. P = 2/4 = 1/2.
Q4. A card is drawn from a standard deck. Find the probability of drawing a Queen. A4. There are 4 Queens. P = 4/52 = 1/13.
Q5. A bag contains 4 red, 3 blue, and 5 green marbles. One marble is drawn at random. Find the probability it is blue. A5. Total = 4 + 3 + 5 = 12. Blue = 3. P(Blue) = 3/12 = 1/4.
Q6. The probability of winning a game is 0.25. Find the probability of losing. A6. P(Losing) = 1 — P(Winning) = 1 — 0.25 = 0.75.
