Class 10 Maths Chapter 12 Exercise 12.1 Solutions 2026 – Probability Notes PDF

Unit 12: Probability | Exercise 12.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 12.1 solutions for Probability have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Measuring How Likely Something Is

Words like “likely,” “unlikely,” or “50-50” describe chance in everyday language, but mathematics needs a precise number. Exercise 12.1 introduces probability as exactly that number — a value between 0 and 1 that measures how likely an event is to happen, built from counting favorable and total outcomes.

Key Concepts Covered in Exercise 12.1

Sample Space and Events

The sample space is the set of every possible outcome of an experiment, such as {1,2,3,4,5,6} for rolling a die. An event is any subset of the sample space — for example, “rolling an even number” is the event {2,4,6}.

The Probability Formula

The probability of an event E is P(E) = (number of favorable outcomes) / (total number of possible outcomes), assuming every outcome is equally likely. This single formula is the foundation for the entire chapter.

The Range of Probability

Probability always lies between 0 and 1 inclusive. P(E) = 0 means the event is impossible, and P(E) = 1 means the event is certain. Everything else falls somewhere in between.

Complementary Events

The complement of an event E (written E′ or ḗ) is everything in the sample space that is not E. Since E and its complement together make up the whole sample space, P(E) + P(E′) = 1.

Step-by-Step Solved Examples

Q. No. 1: A fair die is rolled once. Find the probability of getting a number greater than 4.

Sample space = {1,2,3,4,5,6}, so total outcomes = 6

Favorable outcomes (greater than 4): {5, 6}, so favorable = 2

P(number > 4) = 2/6 = 1/3

Q. No. 2: A bag contains 3 red balls and 5 blue balls. One ball is drawn at random. Find the probability that it is red.

Total balls = 3 + 5 = 8

Favorable outcomes (red) = 3

P(red) = 3/8

Q. No. 3: If the probability that it rains tomorrow is 0.35, find the probability that it does not rain.

P(rain) + P(not rain) = 1

P(not rain) = 1 − 0.35 = 0.65

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 12.1 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on identifying the sample space or total outcomes for a simple experiment
• Short Questions: Direct probability calculations using the favorable/total formula, like Q. No. 1 and 2 above
• Long Questions: Multi-part problems involving complementary events or combined conditions

Where This Matters Beyond This Exercise

The basic probability formula introduced here is used throughout the rest of the chapter, including combined events with two dice or cards in later exercises. Probability itself is used constantly outside the classroom — in weather forecasting, insurance pricing, quality control, and games of chance.

Common Mistakes Students Make in Exercise 12.1

• Miscounting the total number of outcomes in the sample space, especially for compound experiments
• Forgetting to simplify the final probability fraction
• Confusing an event with a single outcome, when an event can include multiple outcomes
• Calculating a probability greater than 1 or less than 0, which is always a sign of a counting error

Why This Exercise Matters for the Board Exam

Basic probability questions are quick, formula-based, and one of the most reliably repeated question types across the whole paper, making Exercise 12.1 a dependable source of marks. Carefully and clearly listing the sample space before calculating often prevents small counting errors that are easy to make under time pressure.

Quick Links – Chapter 12: Probability

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 12.1 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. Why is probability always between 0 and 1?

Probability represents a fraction of favorable outcomes out of all possible outcomes. Since favorable outcomes can never exceed the total, and can be as few as zero, the fraction is always between 0 (impossible) and 1 (certain), inclusive.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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