Class 10 Maths Chapter 12 Exercise 12.3 Solutions 2026 – Probability Notes PDF

Unit 12: Probability | Exercise 12.3 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 12.3 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.

Combining Two Events With “And”

Exercise 12.2 combined two events with “or.” Exercise 12.3 turns to “and” — the probability that two events both happen, one after another or at the same time. This introduces independent events and the multiplication rule, the final major tool in this chapter’s toolkit.

Key Concepts Covered in Exercise 12.3

Independent Events

Two events are independent if the outcome of one has no effect on the probability of the other — for example, tossing a coin and rolling a die are independent, since neither result influences the other.

The Multiplication Rule

For independent events, the probability that both A and B happen is P(A and B) = P(A) × P(B). This is the direct “and” counterpart to the addition rule’s “or.”

Events With and Without Replacement

When drawing multiple items from a group, “with replacement” means the item is put back before the next draw (keeping the probabilities independent), while “without replacement” changes the total each time, meaning the events are no longer independent.

Combining the Addition and Multiplication Rules

Many longer problems in this exercise use both rules together — for instance, using the multiplication rule to find the probability of a specific sequence, then the addition rule to combine multiple such sequences.

Step-by-Step Solved Examples

Q. No. 1: A coin is tossed and a die is rolled. Find the probability of getting heads and a 6.

These events are independent.

P(heads) = 1/2, P(6) = 1/6

P(heads and 6) = 1/2 × 1/6 = 1/12

Q. No. 2: A bag contains 4 red and 6 blue balls. A ball is drawn, its color noted, and replaced. A second ball is then drawn. Find the probability that both balls are red.

Since the first ball is replaced, the draws are independent, and each has the same probabilities.

P(red) = 4/10 = 2/5

P(both red) = 2/5 × 2/5 = 4/25

Q. No. 3: Using the same bag as Q. No. 2 (4 red, 6 blue, total 10), find the probability of drawing two red balls in a row without replacement.

First draw: P(red) = 4/10 = 2/5

Since the first ball isn’t replaced, only 9 balls remain, with 3 red left.

Second draw: P(red | first was red) = 3/9 = 1/3

P(both red, no replacement) = 2/5 × 1/3 = 2/15

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 12.3 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 whether two given events are independent
• Short Questions: Applying the multiplication rule for simple independent events, like Q. No. 1 above
• Long Questions: With-replacement vs. without-replacement problems, like Q. No. 2 and 3 above

Where This Matters Beyond This Exercise

Recognizing whether events are independent, and correctly adjusting probabilities for “without replacement” situations, is essential for the mixed problems in the Review Exercise, which often combine every rule from this chapter in one question. The multiplication rule is also fundamental in genetics (predicting inherited traits), quality control (chance of multiple defects), and cryptography.

Common Mistakes Students Make in Exercise 12.3

• Using the same probability for both draws even when the problem specifies no replacement
• Forgetting that the total number of items decreases by 1 after each draw without replacement
• Confusing independent events (multiplication rule) with mutually exclusive events (addition rule)
• Multiplying probabilities from unrelated sample spaces incorrectly

Why This Exercise Matters for the Board Exam

With-replacement and without-replacement problems are a favorite long-question style in this chapter, since they test whether a student notices the subtle but important difference in wording. Clearly stating whether replacement occurs, and adjusting the second probability accordingly, is often the exact distinction examiners are checking for.

Quick Links – Chapter 12: Probability

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 12.3 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. What is the difference between independent events and mutually exclusive events?

Independent events don’t affect each other’s probability and use the multiplication rule (“and”). Mutually exclusive events cannot happen together and use the addition rule (“or”). These are two entirely different relationships between events, and mixing them up is a common source of errors.

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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