NCERT Solutions Class 9 Maths Exercise 7.1 — The Mathematics of Maybe: Introduction to Probability

NCERT Class 9 Mathematics — Ganita Manjari Part I, The Mathematics of Maybe: Introduction to Probability, Exercise Set 7.1. All four parts solved.

Probability measures how likely something is, on a scale from $0$ to $1$:

$$0 = \text{impossible} \qquad \tfrac12 = \text{even chance} \qquad 1 = \text{certain}$$

This first exercise is about placing events on that scale by judgement, before any calculation is introduced. The five labels the question offers correspond to positions along it:

Key insight. “Impossible” and “certain” are much stronger words than everyday speech suggests. An event is impossible only if it cannot happen — not merely if it never has. So the real work in each part is deciding whether the event is genuinely ruled in or out, or only very likely or very unlikely.

Question 1

Rank the following events on a scale from $0$ (Impossible) to $1$ (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking.

Solution.

(i) The next Monday will come after Sunday

Certain — probability $1$.

This is not a matter of chance at all. The days of the week run in a fixed cycle, and Monday always follows Sunday by definition of the calendar. There is no way it could turn out otherwise, so the probability is exactly $1$.

It is worth noticing why this belongs at the very end of the scale rather than merely near it: nothing needs to go right for it to happen. Events that depend on the world behaving as usual — part (iv), say — can fail in small ways, and so never quite reach $1$.

(ii) It will snow in Mumbai in July

Impossible — probability $0$ (or so close to $0$ as to make no practical difference).

Mumbai is on India’s west coast at sea level, close to the equator. July is the height of the monsoon there, with temperatures around $27$–$30\,°\mathrm{C}$. Snow requires temperatures at or below $0\,°\mathrm{C}$, which is some thirty degrees away. It has never snowed in Mumbai in recorded history.

Strictly speaking, no weather event is logically impossible in the way part (i) is logically certain. But the probability is so small that $0$ is the right place on the scale.

(iii) An elephant will walk through your classroom today

Impossible (or extremely unlikely) — probability $0$, or very close to it.

Elephants are not kept near schools, classrooms have doors far too small for one, and nothing in an ordinary school day brings an elephant anywhere near.

Again this is not logically impossible — a school beside a forest, on the day of a temple procession, could conceivably see one — which is why “extremely unlikely” is the more careful label. But on the $0$–$1$ scale it sits at the very bottom.

(iv) You will greet at least one friend at school tomorrow

More likely — probability close to $1$, but not equal to $1$.

On an ordinary school day you will meet many friends and greet at least one of them almost without thinking. So this is well above the halfway mark.

But it is not certain, and the reasons are worth naming: you might be absent; the school might be closed for a holiday or an emergency; tomorrow might be a Sunday. Each is unlikely, and each is enough to stop the probability reaching $1$.

This is the useful contrast with part (i). Both feel like sure things, but part (i) cannot fail while part (iv) merely almost never does.

Common mistakes

• Part (i), treating it as very likely rather than certain. It is a fact about the calendar, not a prediction about the world. Nothing could make Monday not follow Sunday.
• Parts (ii) and (iii), labelling them “less likely”. “Less likely” means somewhere between $0$ and $\frac12$ — which would include events with a one-in-three chance. These are far more extreme than that, and belong at the bottom of the scale.
• Part (iv), labelling it “certain”. Almost certain is not certain. Naming a way it could fail — being absent, a holiday — is what shows the difference, and is the reasoning the question asks for.
• Confusing “equally likely” with “I don’t know”. Even chance means probability exactly $\frac12$ — as for a fair coin. Being unsure about an event is not the same as its having a fifty-fifty chance.
• Giving the label without the reason. The question asks for reasons, and the reason is what distinguishes a judgement from a guess.

Practise next

• Exercise Set 7.2 — measuring probability rather than judging it, using experimental data and relative frequency.
• The End-of-Chapter Exercises, Questions 1 and 3 — the same scale in question form, and a set of experiments to sort into those with equally likely outcomes and those without.