The Mathematics of Maybe: Introduction to Probability – Class 9 Maths Chapter 7 Notes & NCERT PDF

Introduction

Will it rain today? Will our school win tomorrow's hockey match? Will my name be picked in the monthly lucky draw? We know the possible outcomes of each of these, but not which one will actually happen. Such events are random events — there is an element of chance in them.

Chapter 7 of Ganita Manjari shows that probability is a kind of measurement, just like length, area or volume — except that it measures how likely an event is. We move from guessing ("It's sunny, so rain is unlikely" — a subjective probability) to objective ways of measuring likelihood: by doing experiments, by collecting data, and by reasoning about equally likely outcomes.

Key Ideas

7.1 What is probability?

The probability scale: probability is measured from 0 (impossible) to 1 (certain). Most events lie strictly between 0 and 1.

The book's examples on the scale:

A probability of 0.75 for your school winning means a 75% chance — winning is more likely than not.

7.2 Measuring probability objectively

There are two main objective methods:

Experimental probability = (Number of times the event occurred) ÷ (Total number of trials)

This is also called the relative frequency.

P(Event) = (Number of favourable outcomes) ÷ (Number of possible outcomes)

Statistical probability and sampling: in a class survey of 50 students, 20 like mango, so P(mango) = 20/50 = 0.4. For a school of 1500 students, we would estimate about 40% × 1500 = 600 mangoes. The group asked (50 students) is a sample; the whole school is the population. A larger, more representative sample gives a more trustworthy estimate.

Law of Large Numbers: as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.

Gambler's Fallacy: after six heads in a row, many people feel tails is "due". But a coin has no memory — P(tails) on the next toss is still 1/2. In Snakes and Ladders, after three 6s in a row, P(6) on the next roll is still 1/6. Each roll or toss is independent.

A coin is called fair (unbiased) when it is symmetrical, so neither side is favoured; a random toss means it falls freely without interference.

7.3 Sample spaces and events

- Coin: S = {H, T}, n(S) = 2

- Die: S = {1, 2, 3, 4, 5, 6}, n(S) = 6

- Two coins: S = {HH, HT, TH, TT}, n(S) = 4

- A match: S = {Win, Lose, Draw}

7.4 Tree diagrams

A tree diagram lists all outcomes of a multi-step experiment (e.g. tossing a coin twice). Each path from start to end is one complete outcome. For two tosses the tree gives 4 paths — HH, HT, TH, TT — so P(HH) = 1/4 = 0.25.

Worked Examples

1. Experimental probability. A die is rolled 50 times and shows 4 exactly 8 times.

Experimental probability = 8/50 = 0.16 (16%).

2. Theoretical probability on a die. P(rolling a 4) = 1/6 ≈ 0.167 (16.7%).

3. Letters. A letter is picked at random from PROBABILITY (11 letters, 2 Bs).

P(B) = 2/11 ≈ 0.182 (18.2%).

4. Two coins. P(at least one head): favourable outcomes HH, HT, TH = 3 out of 4, so P = 3/4. P(one head and one tail) = HT, TH = 2/4 = 1/2.

5. Three coins. S has 2 × 2 × 2 = 8 outcomes. Exactly two heads: HHT, HTH, THH → P = 3/8.

6. Using a sample to estimate. A sample of 30 sweets has 10 red, 8 green, 7 yellow, 5 blue.

P(green) = 8/30 = 4/15. In a bag of 600 sweets, expected yellow ≈ (7/30) × 600 = 140.

Figure it Out — Hints

Maths Words

Indian Connection

Did you know? The game of Snakes and Ladders comes from ancient India. It evolved from Jñān-Chaupaḍ, an educational dice game in which each ladder stood for a virtue and each snake for a vice. A 19th-century Jain Jñān-Chaupaḍ on cloth is in the National Museum.

Common Mistakes

Probability in Daily Life

Weather forecasts ("70% chance of rain"), insurance, sales forecasting, sports tosses to decide who bats first, medical research and quality control (like the book's tyre-company data) all use probability. It never tells us what will happen next — it tells us what to expect in the long run.