What is the probability formula?

P(E) = number of favourable outcomes / total number of outcomes. For example, the probability of rolling a 3 on a die is 1/6 — one favourable outcome out of six total.

What is a complementary event?

The complement of event E (written E') is 'not E'. The rule is P(E) + P(E') = 1. For example, if P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7.

What is a sample space?

The sample space (S) is the set of ALL possible outcomes of an experiment. For a die, S = {1, 2, 3, 4, 5, 6}. For two coins, S = {HH, HT, TH, TT}.

Can probability be greater than 1?

No. Probability is always between 0 and 1 inclusive. P = 0 means the event is impossible; P = 1 means the event is certain. Any value outside [0, 1] indicates an error in the calculation.

What is an impossible event?

An impossible event has probability 0 — it can never happen. For example, rolling a 7 on a standard die is an impossible event. The empty set ∅ represents an impossible event.

Chapter 4 · Kerala SSLC Class 10 Maths

Mathematics of Chance

Two ways to learn — a crisp formula reference, or a story that builds probability intuition from scratch.

Sample spaceP(E) formulaComplementary eventsDice & cards

Also useful:Practice Problems — Kerala SSLC Maths 1-on-1 Kerala SSLC Maths Tuition

Kerala SSLCClass 10MathsChapter 4 — Mathematics of Chance⏱ 10 min read

Three real-world scenes: raffle, weather bet, and a dice game.

How this works

Each scene gives you a real-world situation — a raffle, a weather forecast, a dice game — and builds the probability formula naturally from it. Follow the story, then open the maths box for the formal rule.

The Raffle Ticket

It is Sports Day at school. The teacher announces a raffle — 100 tickets have been sold, and one lucky winner gets a prize. You bought exactly 1 ticket. Your friend bought 10 tickets.

You

What are my chances of winning? I only have one ticket out of a hundred.

Think about it simply. There are 100 tickets in the bag. Only 1 of them is yours. If the draw is random and fair, every ticket has the same chance of being picked.

Your tickets = 1

Total tickets = 100

P(you win) = 1/100 = 0.01

Your friend, with 10 tickets, has a better shot:

Friend's tickets = 10

Total tickets = 100

P(friend wins) = 10/100 = 1/10 = 0.1

This is the heart of probability. You are counting how many ways your event can happen and dividing by the total number of equally likely outcomes. The full set of all outcomes (all 100 tickets) is called the sample space.

The sample space S = {ticket 1, ticket 2, …, ticket 100}. The event E = { your ticket }. P(E) = n(E)/n(S) = 1/100.

The Weather Bet

The morning weather report says: "There is a 60% chance of rain today." Your friend Arjun bets it will rain. You bet it will not rain. Who has the better bet?

Arjun

The forecast said 60% chance of rain — that means P(rain) = 0.6. I like my odds.

You think for a moment. If P(rain) = 0.6, what is P(no rain)?

There are only two outcomes: either it rains, or it does not. Together they cover everything — the full sample space. So their probabilities must add up to 1:

P(rain) + P(no rain) = 1

0.6 + P(no rain) = 1

P(no rain) = 1 − 0.6 = 0.4

You

So P(no rain) = 0.4. Arjun has the better bet — 0.6 vs 0.4. I lose the bet but learn the rule!

The event "no rain" is called the complement of the event "rain." Written as E' (read "E prime" or "E complement"). Their probabilities always sum to exactly 1 — because together they cover every possible outcome.

The Dice Game

At a board game night, you are playing a game where two dice are rolled and you win if the sum is 7. Your friend says 7 is the luckiest number. But why is 7 the most likely sum?

Friend

I think every sum from 2 to 12 is equally likely. There are 11 possible sums, so each has a 1/11 chance, right?

Actually — no. This is the most common mistake with two dice. The sample space is not the 11 sums. Each die has 6 faces, so there are 6 × 6 = 36 ordered pairs:

Sample Space — Two Dice (36 outcomes)

(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

The highlighted pairs are those that sum to 7. Count them: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — that is 6 outcomes.

n(S) = 36 (all ordered pairs)

n(sum=7) = 6 (pairs summing to 7)

P(sum = 7) = 6/36 = 1/6

No other sum has 6 pairs. That is why 7 is the most likely. A sum of 2 only has 1 way: (1,1). A sum of 12 only has 1 way: (6,6).

The Big Picture

Every probability question in this chapter reduces to three steps: list the sample space completely, count the favourable outcomes, divide. The formula never changes — only the sample space does.

Find probability of event E?

P(E) = n(E) / n(S)

Find P(not E)?

P(E') = 1 − P(E)

Two dice — how many outcomes?

n(S) = 36

Check if probability is valid?

0 ≤ P(E) ≤ 1

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