What is Probability? — Core Concepts for Exercise 13.2

Probability is the branch of mathematics that measures the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Exercise 13.2 of Class 10 Mathematics (CBSE, Telangana, and Andhra Pradesh syllabus) covers classical (theoretical) probability — calculating probabilities by counting equally likely outcomes without actually performing the experiment.

Three essential terms you must know before solving any problem in this exercise:

• Sample Space (S) — the set of all possible outcomes of an experiment. For example, rolling one die gives S = {1, 2, 3, 4, 5, 6}, so n(S) = 6.
• Event (A) — any specific outcome or group of outcomes you are interested in. The number of outcomes in that event is written as n(A).
• Complementary Event (Ā) — the event that A does NOT happen. P(Ā) = 1 − P(A). If the probability of picking a red ball is 3/8, the probability of NOT picking a red ball is 1 − 3/8 = 5/8.

Question 1 — Red and Black Balls in a Bag

Question 2 — Red, White and Green Marbles

Question 3 — Coins Falling from a Kiddy Bank

Question 4 — Fish from an Aquarium Tank

Question 5 — The Spinning Arrow (Numbers 1 to 8)

A spinning arrow can point to any of the numbers 1, 2, 3, 4, 5, 6, 7, 8 with equal probability. The sample space has n(S) = 8 equally likely outcomes.

Question 6 — A Deck of 52 Playing Cards

This is one of the most important and most-examined topics in Probability for Class 10 board exams. You must know the structure of a standard deck of 52 cards perfectly. Here is a complete breakdown:

Question 7 — Five Diamond Cards (Without Replacement)

This question introduces an important concept: probability without replacement. When a card is removed and NOT put back, the sample space for the next draw changes.

Questions 8 & 9 — Defective Pens and Bulbs

Question 10 — Numbered Discs from 1 to 90

Question 11 — Geometric Probability (Rectangle and Circle)

This question introduces geometric probability — where the probability is the ratio of areas rather than the count of outcomes. This is a unique type that frequently appears in board exams.

Question 12 — Sudha Buying a Pen

Question 13 — Rolling Two Dice: Sum Probability Table

When two dice are rolled simultaneously, the total number of outcomes is 6 × 6 = 36. The possible sums range from 2 (both dice show 1) to 12 (both dice show 6). The number of ways to achieve each sum follows a symmetric pattern — increasing from sum 2 to sum 7, then decreasing back.

Question 14 — Three Coin Tosses: Deekshitha's Game

When 3 coins are tossed, the total outcomes = 2³ = 8. Deekshitha wins only if all three tosses match (HHH or TTT). Otherwise she loses.

Question 15 — A Die Thrown Twice: Getting (or Not Getting) 5

Throwing a die twice is the same experiment as rolling two dice simultaneously — the sample space has n(S) = 36 equally likely outcomes.

Quick Reference — All Answers at a Glance

Common Mistakes to Avoid in Board Exams

• Forgetting the "without replacement" update: In Q7 and Q9, once a card or bulb is removed without being put back, both n(S) and n(A) must be recalculated. Students who forget this score zero on the second part.
• Wrong count of face cards: There are 12 face cards (3 per suit × 4 suits = J, Q, K of each suit), NOT 16. Ace is NOT a face card in standard probability convention.
• Confusing "at least once" with "exactly once": "At least once" = 1 or more times. Always use the complementary approach: P(at least once) = 1 − P(never). This avoids long enumeration.
• Assuming 11 dice-sum outcomes are equally likely (Q13 trap): They are NOT. The 36 ordered pairs are the equally likely outcomes. Sums are derived values with different frequencies.
• Not simplifying fractions: Board examiners expect fractions in their lowest terms. 100/180 must be simplified to 5/9. 132/144 must be simplified to 11/12.
• Using diameter instead of radius for circle area: In Q11, diameter = 1m, so radius = 0.5m. Area = π(0.5)² = π/4. Using diameter directly in the formula will give a wrong answer.

What This Exercise Prepares You For

Exercise 13.2 covers the bulk of the marks-earning problems in Chapter 13. The concepts here — complementary events, without-replacement problems, deck of cards, and two-dice outcomes — appear in nearly every Telangana, Andhra Pradesh, and CBSE Class 10 board paper in the 2–4 mark question section.

Mastering this exercise also builds intuition for the Exercise 13.1 (Introduction to Probability) concepts, and creates a strong foundation for advanced probability topics encountered in Class 11 and 12 (such as conditional probability and Bayes' theorem). Students preparing for competitive exams like JEE Foundation will find the two-dice and cards problems particularly relevant.