Probability Class 10 ICSE — Bansal Chapter 25
Probability Class 10 ICSE introduces the mathematics of chance. Everyday phrases like ‘most likely’ and ‘no chance at all’ all carry uncertainty. This chapter, the 25th in the Bansal maths book, turns that uncertainty into numbers you can calculate. It defines the key terms, gives the probability formula, and shows you how to count outcomes for coins, dice and playing cards.
Download Probability Class 10 ICSE Mathematics Chapter 25 PDF
Download the Probability Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 16 pages, 4 sections, 63 questions.
What is inside Chapter 25
16 pages · 4 sections · 9 worked examples · 63 exercise questions
| Section | Title |
|---|---|
| 25.1 | Introduction |
| 25.2 | Some Basic Terms and Concepts |
| 25.3 | Measurement of Probability |
| 25.4 | Important |
- EXERCISE 25(A): 15 questions
- EXERCISE 25(B): 14 questions
- EXERCISE 25(C): 34 questions
25.1 Introduction
Everyday speech is full of uncertainty. People say ‘most likely’, ‘almost uncertain’ or ‘no chance at all’. Read them carefully, and each one carries an element of uncertainty. The measure of that uncertainty is the theory of probability.
The book says the study of probability carries both mathematical interest and practical importance. It turns the unknown into a number you can calculate. In the book’s chapter order, probability comes right after the chapter on the cone and sphere, which measured solid shapes. This chapter measures chance instead.
No formula appears yet in this section. Section 25.2 builds the vocabulary the formula needs: experiment, random experiment, sample space, equally likely outcomes and event. Section 25.3 introduces the ratio P(E) = m/n. Section 25.4 gives the counting shortcuts for coins, dice and cards.
25.2 Some Basic Terms and Concepts
This section defines five working terms. Every question in the chapter uses them.
- Experiment — a process with a well-defined outcome, like tossing a coin.
- Random experiment — all outcomes are known in advance, but the specific outcome is not.
- Sample space (S) — the set of all possible outcomes of an experiment.
- Equally likely outcomes — each outcome has the same chance as any other.
- Event — an outcome of a random experiment, something that happens.
Sample spaces are the counting lists used later. One coin gives S = \{H, T\}. One die gives S = \{1, 2, 3, 4, 5, 6\}. Two coins give S = \{(H,H), (H,T), (T,H), (T,T)\}.
Two dice give 36 ordered pairs, from (1,1) to (6,6). A coin and a die together give 12 pairs, from (H,1) to (T,6). Order matters in these lists. When two dice are thrown, (3,5) and (5,3) are different outcomes because the dice are separate objects.
Equally likely outcomes need checking. Consider a bag with 6 red and 2 yellow balls. A single draw gives red or yellow, but red is much more likely. Those outcomes are not equally likely. If the bag held 4 red and 4 yellow, the two outcomes would be equally likely, each with probability \frac{1}{2}.
Drawing a card from a deck gives 52 events, one for each card. That idea — one outcome per card — is what the card questions build on.
25.3 Measurement of Probability
The probability of an event measures how likely it is to happen. If a random experiment has n total outcomes and m of them are favourable to event E, then:
P(E) = \frac{m}{n} = \frac{\text{Number of outcomes favourable to E}}{\text{Total number of possible outcomes}}
The ratio works because each outcome is assumed equally likely. Event E’s probability is its share of the whole outcome set. An event with exactly one favourable outcome, like getting a head, is an elementary event. This is the classical definition, and the chapter uses it throughout. Later chapters in the Class 10 ICSE maths book carry the same careful counting into statistics.
Two kinds of probability appear in the book. Empirical probability is based on an actual experiment. Classical probability is calculated without repeating the experiment.
| Feature | Empirical probability | Classical probability |
|---|---|---|
| Source | Actual experiment | Reasoning, no experiment needed |
| Assumes equally likely outcomes | No | Yes |
| Result | An estimate that can change | An exact value |
| Example | 57 heads in 100 tosses gives \frac{57}{100} | One toss gives \frac{1}{2} for a head |
Complementary events. If E is an event, the event that E does not happen is its complement, written \overline{E}. The two cover every possible outcome, so:
P(E) + P(\overline{E}) = 1 \Rightarrow P(\overline{E}) = 1 - P(E)
Watch the phrasing in die questions. ‘Not greater than 2’ and ‘less than or equal to 2’ mean the same thing.
Worked pattern with numbers not taken from the textbook. A bag holds 5 red, 3 blue and 2 green balls.
Step 1: Count the total outcomes. The bag holds 5 + 3 + 2 = 10 balls.
Step 2: Count the favourable outcomes for each colour.
P(\text{red}) = \frac{5}{10} = \frac{1}{2}, \quad P(\text{blue}) = \frac{3}{10}, \quad P(\text{green}) = \frac{2}{10} = \frac{1}{5}
Step 3: Check that all outcomes add to 1.
\frac{1}{2} + \frac{3}{10} + \frac{1}{5} = \frac{5 + 3 + 2}{10} = 1
Step 4: Use the complement for ‘not red’.
P(\text{not red}) = 1 - P(\text{red}) = 1 - \frac{1}{2} = \frac{1}{2}
Final answer: Red \frac{1}{2}, blue \frac{3}{10}, green \frac{1}{5}, not red \frac{1}{2}.
Extremes exist too. Getting a 7 on a single die has probability \frac{0}{6} = 0, so it is an impossible event. Getting a number less than 7 has probability \frac{6}{6} = 1, so it is a sure event. For any event E, 0 \leq P(E) \leq 1.
The face card example uses the same formula. A deck has 12 face cards, so one draw gives P(\text{face card}) = \frac{12}{52} = \frac{3}{13}. By the complement rule, P(\text{not face card}) = 1 - \frac{3}{13} = \frac{10}{13}.
25.4 Important
This section gives counting shortcuts. One coin has 2^1 = 2 outcomes. Two coins have 2^2 = 4. Three coins have 2^3 = 8. In general, n coins give 2^n outcomes.
Dice follow the same pattern with base 6. One die gives 6^1 = 6 outcomes. Two dice give 6^2 = 36. In general, n dice give 6^n outcomes.
The wording changes, the outcomes do not. Tossing one coin twice gives the same outcomes as tossing two coins together. Rolling one die twice gives the same outcomes as rolling two dice together.
Worked pattern: two dice are thrown, and you want the sum to be 7.
Step 1: Total outcomes for two dice are 6^2 = 36.
Step 2: List the pairs that add to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Count = 6.
P(\text{sum } 7) = \frac{6}{36} = \frac{1}{6}
Final answer: \frac{1}{6}.
Three coins give eight outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. The book’s table counts the interesting events:
| Event | Favourable outcomes | Count |
|---|---|---|
| All heads | HHH | 1 |
| Exactly two heads | HHT, HTH, THH | 3 |
| Exactly one head | HTT, THT, TTH | 3 |
| At least one head | All except TTT | 7 |
| At least two heads | HHT, HTH, THH, HHH | 4 |
| All tails | TTT | 1 |
Memory aid for the 52-card deck. A deck has 4 suits of 13 cards each, so 4 \times 13 = 52. Kings, queens and jacks are the face cards: 4 \times 3 = 12 in all. The deck splits into 26 red and 26 black cards.
Common mistakes to avoid when counting outcomes for two dice or two coins:
- Treating (3,5) and (5,3) as one outcome. They are two different outcomes.
- Using 6 as the total for two dice instead of 36.
- Forgetting that ‘at least one head’ with three coins includes HHH, giving 7 outcomes, not 3.
What to practise from Probability Class 10 ICSE
The chapter ends with three exercise blocks. All of them use the same formula, so practise counting the outcomes first.
- EXERCISE 25(A) — 15 questions on the basic formula, complementary events, and single coin or die throws.
- EXERCISE 25(B) — 14 questions on cards, numbered cards, coins and two-dice problems.
- EXERCISE 25(C) — 34 mixed questions, several tagged with board exam years.
Across exercises 25(B) and 25(C), the textbook tags these questions with board exam years:
| Board year tag | Question | Pattern it tests |
|---|---|---|
| [2010] | 25(C) Q29 | Cards numbered 1 to 20: prime, divisible by 3, perfect square |
| [2011] | 25(C) Q32 | Multiples of 3 removed from a pack; drawing a face card or an even red card |
| [2013] | 25(C) Q31 | Solving for the number of black balls when that probability is two-fifths of the white ball’s |
| [2014] | 25(C) Q33 | A die with positive and negative numbers on its faces |
| [2015] | 25(C) Q34 | A bag with white, red and green balls |
These tagged questions show the patterns the board has used before. Work through 25(C) Q31 and Q34 carefully; both combine probability with equation solving or careful counting.
You can verify the board’s syllabus details on the official CISCE website. For the chapters before and after this one, see the Class 10 ICSE books index.
Probability Class 10 ICSE frequently asked questions
What is the probability formula in Class 10 ICSE Maths?
The formula is P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}. Count the total outcomes first, then count how many match the event. Every exercise in the chapter uses this ratio.
How many outcomes are there when two dice are thrown in Probability Class 10 ICSE?
There are 36. Each die has 6 faces, so 6 \times 6 = 36 ordered pairs. The pairs (1,3) and (3,1) are counted as different outcomes.
What is the difference between empirical and classical probability in this ICSE chapter?
Empirical probability comes from an actual experiment, like 57 heads in 100 tosses. Classical probability assumes equally likely outcomes and gives the exact value without repeating the experiment. This chapter uses classical probability.
What does P(not E) = 1 – P(E) mean for Class 10 ICSE probability questions?
An event and its complement cover every possible outcome, so their probabilities add to 1. If P(E) = \frac{3}{13}, then P(\text{not } E) = 1 - \frac{3}{13} = \frac{10}{13}.
How many face cards are in a deck of 52 cards in this ICSE chapter?
There are 12 face cards. Each suit has a king, queen and jack, and there are four suits, so 4 \times 3 = 12. The probability of drawing a face card is \frac{12}{52} = \frac{3}{13}.
Reference: ICSE Class 10 Mathematics textbook material, chapter 25: Probability — CISCE-aligned.
More from this book
Explore Class 10 Maths Notes
Related chapters: