Class 10 Maths Chapter 12 Exercise 12.2 Solutions 2026 – Probability Notes PDF

Unit 12: Probability | Exercise 12.2 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 12.2 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 “Or”

Exercise 12.1 calculated the probability of a single event. Exercise 12.2 asks a natural follow-up question: what is the probability that at least one of two events happens? The answer depends on whether the two events can ever happen at the same time, which is exactly what the addition rule accounts for.

Key Concepts Covered in Exercise 12.2

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both happen at the same time — for example, rolling a 2 and rolling a 5 on the same die roll. For mutually exclusive events, P(A or B) = P(A) + P(B).

The General Addition Rule

When two events can overlap (are not mutually exclusive), simply adding their probabilities would double-count the overlap. The general addition rule corrects for this: P(A or B) = P(A) + P(B) − P(A and B).

Identifying the Overlap

The key skill in this exercise is recognizing P(A and B) — the probability that both events happen together — and subtracting it exactly once so it isn’t counted twice.

Choosing the Correct Rule

Before applying either version of the addition rule, this exercise gives practice first deciding whether two given events are mutually exclusive or overlapping, since that decision determines which formula to use.

Step-by-Step Solved Examples

Q. No. 1: A die is rolled once. Find the probability of getting a 2 or a 5.

These events are mutually exclusive (a single roll can’t show both 2 and 5).

P(2) = 1/6, P(5) = 1/6

P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3

Q. No. 2: A card is drawn from a standard deck of 52 cards. Find the probability that it is a heart or a king.

P(heart) = 13/52, P(king) = 4/52

These events overlap: the king of hearts is both a heart and a king, so P(heart and king) = 1/52

P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13

Q. No. 3: In a class of 40 students, 18 play cricket, 15 play football, and 6 play both. Find the probability that a randomly chosen student plays cricket or football.

P(cricket) = 18/40, P(football) = 15/40, P(cricket and football) = 6/40

P(cricket or football) = 18/40 + 15/40 − 6/40 = 27/40

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 12.2 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 mutually exclusive
• Short Questions: Applying the simple addition rule for mutually exclusive events, like Q. No. 1 above
• Long Questions: Applying the general addition rule with an overlap, like Q. No. 2 and 3 above

Where This Matters Beyond This Exercise

The distinction between mutually exclusive and overlapping events reappears in the next exercise when combining probabilities of independent events with “and.” This same addition rule is used constantly in fields like insurance and risk assessment, wherever the chance of “at least one” of several possible problems occurring needs to be calculated.

Common Mistakes Students Make in Exercise 12.2

• Forgetting to subtract the overlap when two events are not mutually exclusive
• Assuming two events are mutually exclusive without checking whether they can happen together
• Double-subtracting the overlap by mistake, instead of subtracting it exactly once
• Adding probabilities from different sample spaces without a common total

Why This Exercise Matters for the Board Exam

Addition rule questions are a compact way for examiners to test whether students correctly distinguish between mutually exclusive and overlapping events — a conceptual distinction that’s easy to get wrong under time pressure. Explicitly stating which case applies before writing the formula tends to earn a method mark even before any calculation begins.

Quick Links – Chapter 12: Probability

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Download Exercise 12.2 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 12.2 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. How do I know if two events are mutually exclusive?

Two events are mutually exclusive if they cannot both occur at the same time as a result of a single trial — for example, a single card cannot be both a heart and a spade. If an outcome could satisfy both events at once, they are not mutually exclusive, and the overlap must be subtracted.

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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