ICSE Class 10 Mathematics Probability: Quick Guide for Board Exams

Why Probability Pops Up Everywhere

Ever wondered why we talk about the chance of rain or the odds of winning a game? That’s probability in action – and it shows up a lot in your ICSE maths paper.

💡 In Simple Words: Probability tells you how likely something is to happen. If you have 4 red marbles and 1 blue marble in a bag, the chance of pulling out a red one is 4 out of 5, or 0.8.

What is Probability? A Friendly Definition

Probability is a number between 0 and 1 that measures how often an event would occur if you repeated an experiment many times. Zero means it never happens, one means it always happens.

Key Terms You Must Know

• Experiment: Any action you can repeat, like tossing a coin or drawing a card.
• Sample Space: The complete set of all possible outcomes of an experiment. Think of it as a list of every result you could possibly get.
• Event: A specific outcome or a group of outcomes you care about. For example, getting a head when you flip a coin.
• Favourable Outcomes: The outcomes that make the event happen.
• Probability Formula: P(event) = (Number of favourable outcomes) / (Total number of outcomes in the sample space).

How to Solve Basic Probability Problems

Follow these three steps every time you see a probability question:

1. Identify the experiment and write down the sample space.
2. Count the favourable outcomes for the event you’re asked about.
3. Plug the numbers into the probability formula.

Worked Example 1 – Coin Toss

Question: A fair coin is tossed once. What is the probability of getting a head?

Solution:

• Experiment: Toss a coin.
• Sample Space: {Head, Tail} – two possible outcomes.
• Favourable Outcomes: {Head} – only one outcome gives a head.
• Using the formula: P(Head) = 1 / 2 = 0.5.

So there’s a 50% chance of seeing a head.

Worked Example 2 – Drawing a Card

Question: From a standard deck of 52 cards, what is the probability of drawing a queen?

Solution:

• Experiment: Pick one card without looking.
• Sample Space: 52 cards (each card is a possible outcome).
• Favourable Outcomes: 4 queens (one in each suit).
• P(Queen) = 4 / 52 = 1 / 13 ≈ 0.0769.

That’s about a 7.7% chance.

Worked Example 3 – Rolling Two Dice

Question: Two fair dice are rolled together. What is the probability that the sum of the numbers is 7?

Solution:

• Experiment: Roll two dice.
• Sample Space: 6 × 6 = 36 possible ordered pairs (1‑1, 1‑2, …, 6‑6).
• Favourable Outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – six pairs give a sum of 7.
• P(Sum = 7) = 6 / 36 = 1 / 6 ≈ 0.1667.

So you have roughly a 16.7% chance of the dice adding up to seven.

Common Tricks for Board Exams

These shortcuts save you time when the exam clock is ticking:

• Complementary Probability: Instead of counting unfavourable outcomes, subtract the probability of the opposite event from 1. Example: P(not getting a six on a die) = 1 – P(6) = 1 – 1/6 = 5/6.
• Symmetry: If the problem is symmetric (like a fair coin), you can often guess the answer is ½ without writing out the sample space.
• Reducing Fractions Early: Simplify the fraction as soon as you can to avoid big numbers later.

Quick Reference Table

Situation	Formula / Tip	Example
Single event	P = favourable / total	Coin toss – P(Head) = 1/2
Complement	P(not A) = 1 – P(A)	Not getting a 3 on a die = 1 – 1/6 = 5/6
Two independent events (both must happen)	P(A and B) = P(A) × P(B)	Two heads in a row = ½ × ½ = ¼
Two mutually exclusive events (either one happens)	P(A or B) = P(A) + P(B)	Rolling a 1 or a 2 on a die = 1/6 + 1/6 = 1/3

Practice Problems to Test Yourself

1. A bag contains 3 red, 2 blue, and 5 green balls. If one ball is drawn at random, what is the probability of getting a blue ball?
2. A die is rolled twice. Find the probability that the first roll shows an even number and the second roll shows a 5.
3. From a set of letters {A, B, C, D, E}, two letters are picked without replacement. What is the probability that both are vowels? (Assume only A and E are vowels.)

📝 Likely Exam Questions

1. Question: A box contains 4 white, 3 black, and 2 red marbles. One marble is drawn at random. Find the probability of drawing a black marble.
   Answer: Total marbles = 9. Favourable = 3 black. P = 3/9 = 1/3.
2. Question: Two fair coins are tossed together. What is the probability of getting exactly one head?
   Answer: Sample space = {HH, HT, TH, TT}. Favourable = {HT, TH} → 2 outcomes. P = 2/4 = 1/2.
3. Question: A student picks a card at random from a deck of 52 cards. What is the probability that the card is either a heart or a king?
   Answer: Hearts = 13 cards. Kings = 4 cards, but the King of hearts is counted twice, so subtract one. Total favourable = 13 + 4 – 1 = 16. P = 16/52 = 4/13.
4. Question: A die is rolled once. Find the probability of getting a number greater than 4.
   Answer: Numbers greater than 4 are {5,6} → 2 outcomes. P = 2/6 = 1/3.
5. Question: From a set of 7 different books, two are chosen at random. What is the probability that both chosen books are mathematics books if exactly 3 of the 7 are mathematics books?
   Answer: Total ways to choose 2 books = C(7,2) = 21. Favourable ways = C(3,2) = 3. P = 3/21 = 1/7.