Ever wondered why a tossed coin lands heads about half the time? That tiny mystery is what probability is all about.
💡 In Simple Words: Probability tells us how likely something is to happen. If you have 2 red marbles and 2 blue marbles in a bag, the chance of pulling a red one is 2 out of 4, or ½.
What is probability in everyday language?
Think of probability as a way to measure uncertainty. It’s a number between 0 (impossible) and 1 (certain). We usually write it as a fraction, a decimal, or a percent.
Key words you’ll see
- Event: Anything you want to find the chance of, like "getting a head".
- Sample space: The full list of all possible outcomes. Imagine a drawer with every colour of ball you could possibly draw.
- Favourable outcomes: The outcomes that make the event happen.
How to calculate basic probability
The core formula is simple:
Probability = (Number of favourable outcomes) ÷ (Total number of outcomes in the sample space)
Let’s walk through a couple of classic ICSE‑style problems.
Example 1: Rolling a die
Find the probability of getting an even number when you roll a fair six‑sided die.
Sample space = {1,2,3,4,5,6} (six possible results). Favourable outcomes = {2,4,6} (three even numbers). So:
Probability = 3/6 = 1/2 = 0.5 = 50%.
Example 2: Picking a card
A standard deck has 52 cards, 13 of each suit. What’s the chance of drawing a heart?
Favourable outcomes = 13 hearts. Total outcomes = 52 cards.
Probability = 13/52 = 1/4 = 0.25 = 25%.
Common exam‑type problems
ICSE exams love a mix of straight‑forward calculations and a little twist. Here are three patterns you’ll meet.
1. “At least one” problems
Question: Two coins are tossed. What’s the probability of getting at least one head?
Instead of listing all cases, use the complement rule: Probability of at least one head = 1 – probability of no heads (i.e., both tails).
Both tails probability = 1/4, so answer = 1 – 1/4 = 3/4.
2. “Without replacement” problems
Question: A bag contains 3 red and 2 blue balls. Two balls are drawn one after another without putting the first back. Find the probability both are red.
First draw red: 3/5. After removing one red, remaining reds = 2, total balls = 4. Second draw red: 2/4 = 1/2. Multiply because both events must happen: (3/5) × (1/2) = 3/10.
3. “With replacement” problems
Same bag, but you put the first ball back before the second draw. Probability both are red?
Each draw is independent, so simply (3/5) × (3/5) = 9/25.
Quick reference table
| Problem Type | Key Idea | Formula |
|---|---|---|
| Single event | Count favourable ÷ total | P = f / n |
| At least one | Use complement | P = 1 – P(no event) |
| Without replacement | Multiply successive fractions, reduce total each step | P = (f1/n1) × (f2/n2)… |
| With replacement | Multiply same fraction each time | P = (f/n)^k |
Tips to ace ICSE probability questions
- Always write the sample space first – it prevents silly mistakes.
- Check if the draws are with or without replacement; that changes independence.
- When asked for “at least” or “at most”, think complement.
- Convert your final answer to the form the question asks – fraction, decimal, or percent.
- Practice the classic dice, cards, and ball‑bag setups; they reappear every year.
📝 Likely Exam Questions
- Two dice are rolled. Find the probability that the sum is 7.
Answer: Favourable pairs = (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 ways. Total outcomes = 36. Probability = 6/36 = 1/6. - A box contains 4 green, 5 yellow and 1 red marble. One marble is drawn. What is the probability of not getting a red marble?
Answer: Non‑red marbles = 9. Total = 10. P = 9/10. - Three cards are drawn from a deck of 52 without replacement. Find the probability that all three are spades.
Answer: (13/52) × (12/51) × (11/50) = 1716/132600 ≈ 0.0129 ≈ 1.29%. - From a bag of 6 white and 4 black balls, two balls are drawn with replacement. What is the probability that both are white?
Answer: (6/10) × (6/10) = 36/100 = 9/25. - A coin is tossed three times. What is the probability of getting exactly two heads?
Answer: Number of ways to choose 2 heads out of 3 tosses = C(3,2)=3. Each specific sequence has probability (1/2)^3 = 1/8. So P = 3 × 1/8 = 3/8.