NCERT Solutions for Class 9th Maths Chapter 7 The Mathematics of Maybe: Introduction to Probability
Updated on 2026-09-19
About this chapter
Probability measures how likely an event is , on a scale from 0 to 1. For an event E, 0 ≤ P(E) ≤ 1. P(E) = 0 means impossible, P(E) = 1 means certain, P(E) = 1/2 means an even chance. Nothing outside [0, 1] is a probability. A random experiment is one you can repeat, whose full list of possible results you know in advance, but whose result on any single trial you cannot know. The complete list of possible results is the sample space S ; each item in it is an outcome ; the number of items is the sample size n(S) . An event is any subset of S — one outcome, several outcomes, or none. Experimental probability = (number of times the event occurred) ÷ (total number of trials). It is a relative frequency — it describes the data you have, so different sets of trials give different values. Theoret
- What is Probability?
- What is Randomness?
- The Probability Scale
- Analysing Statistical Data Using Probability
- Measuring Probability Objectively
- Sample Space
- Elements of Probability: Sample Spaces and Events
- Tree diagrams
- Chapter 7 review (questions marked * are the harder set)
Quick revision
| Idea | In symbols / formula | What it really says | Where students slip |
|---|---|---|---|
| Outcome | one element of S | A single, complete result of one trial | Calling a group of results one outcome |
| Sample space | S = {o1, o2, …, on} | Every possible outcome, each listed exactly once | Leaving out an outcome, or listing one twice |
| Sample size | n(S) | How many outcomes S contains | Counting types instead of outcomes |
| Event | E ⊆ S | Any selection of outcomes from S | Forgetting that E may hold 0, 1 or many outcomes |
| Experimental probability | (times event occurred)/(total trials) | A relative frequency measured from data | Expecting it to equal the theoretical value exactly |
| Theoretical probability | P(A) = n(A)/n(S) | Fraction of the equally likely outcomes that are favourable | Using it when the outcomes are not equally likely |
| Probability scale | 0 ≤ P(E) ≤ 1 | 0 impossible · 1/2 even chance · 1 certain | Writing an answer bigger than 1, or a negative one |
| Complement | P(not E) = 1 − P(E) | Every trial either gives E or does not | Adding when you should subtract |
| Two-stage tree | P(path) = product of its branches | The branches on one path happen one after the other | Multiplying when you should add, and the reverse |
| Independence | P stays the same whatever happened before | A fair die has no memory | Gambler's Fallacy: thinking a 6 is now 'due' |
| Without replacement | second denominator drops by 1 | The first draw really has changed the box | Reusing the first-stage fractions at the second stage |
| Law of Large Numbers | experimental → theoretical | More trials, steadier relative frequency | Reading it as a promise about the next trial |
Exercises
- In-text Questions — What is Probability? Page 155
- Think and Reflect — What is Randomness? Page 156
- In-text Questions — What is Randomness? Page 157
- Think and Reflect — What is Randomness? Page 157
- Exercise Set 7.1 — The Probability Scale Page 159
- Think and Reflect — Analysing Statistical Data Using Probability Page 163
- Exercise Set 7.2 — Measuring Probability Objectively Page 165–166
- Think and Reflect — Sample Space Page 167
- Exercise Set 7.3 — Elements of Probability: Sample Spaces and Events Page 167–168
- Think and Reflect — Tree diagrams Page 169
- Exercise Set 7.4 — Tree diagrams Page 169
- Chapter 7 review (questions marked * are the harder set) — End-of-Chapter Exercises Page 169–173