7 min readConcept sheet1-mark MCQ; the definition with an example is a 2-mark question

Probability: the classical definition

When all outcomes are equally likely, P(E) = favourable outcomes ÷ total outcomes. It is a number from 0 to 1.

Classical definitionP(E) = number of outcomes favourable to E ÷ number of all possible outcomes

ConditionAll the outcomes must be equally likely

Elementary eventAn event with only one outcome; the probabilities of all elementary events add up to 1

Range0 ≤ P(E) ≤ 1

Words you need

An experiment is an action whose result is not certain, like tossing a coin. Each possible result is an outcome. Outcomes are equally likely when none is favoured: a fair die shows 1 to 6 with the same chance. An event is a set of outcomes, like 'getting an even number'. The outcomes that make the event happen are the favourable outcomes.

Experiment, outcomes and event

Experiment	All outcomes	Event	Favourable outcomes	P(E)
Toss a coin	H, T (2)	Getting a head	H (1)	1/2
Throw a die	1, 2, 3, 4, 5, 6 (6)	Getting an even number	2, 4, 6 (3)	3/6 = 1/2
Pick a day of the week at random	Monday to Sunday (7)	A day starting with T	Tuesday, Thursday (2)	2/7

Worked example 1: prime numbers

Problem

A number is chosen at random from 1 to 20. Find the probability that it is a prime number.

Working

Formula: P(E) = favourable outcomes ÷ total outcomes.
Total outcomes = 20 (the numbers 1 to 20).
Primes from 1 to 20: 2, 3, 5, 7, 11, 13, 17, 19. That is 8 favourable outcomes. Note that 1 is not a prime.
P(prime) = 8/20 = 2/5.

Answer

P(prime) = 2/5.

Worked example 2: a Diwali lucky draw

Problem

In a Diwali lucky draw at a housing society, tickets are numbered 1 to 30. One ticket is drawn at random. Find the probability that its number is a multiple of 4.

Working

Formula: P(E) = favourable outcomes ÷ total outcomes.
Total outcomes = 30.
Multiples of 4 up to 30: 4, 8, 12, 16, 20, 24, 28. That is 7 outcomes.
P(multiple of 4) = 7/30.

Answer

P(multiple of 4) = 7/30.

Equally likely or not?

Equally likelyA fair coin, a fair die, a card from a well-shuffled pack, a ball drawn at random. Each outcome has the same chance.

Not equally likelyWill it rain today or not? Will a team win or lose? These two outcomes need not have the same chance, so you cannot just say 1/2.

The classical formula works only for equally likely outcomes. Words like 'fair', 'at random' and 'well-shuffled' tell you this.

In real life

A lucky draw at a school fete, the toss before a cricket match and picking a chit for your turn all use equally likely outcomes. Probability tells you the fair chance of each.

Exam angle

1 marks: “A letter is chosen at random from the word INDIA. Find the probability that it is I.” The word has 5 letters and 2 of them are I. P = 2/5.

2 marks: “Define the probability of an event E. A number is picked at random from 1 to 25. Find the probability that it is a perfect square.” P(E) = favourable outcomes ÷ total outcomes, when all outcomes are equally likely. Squares: 1, 4, 9, 16, 25 (5). P = 5/25 = 1/5.

2 marks: “Why is the probability of a head on tossing a fair coin 1/2, while the probability that a cricket team wins a match is not 1/2?” A fair coin's two outcomes are equally likely. Winning and losing depend on skill and conditions, so they are not equally likely.

Remember it

Probability is 'part over whole': favourable outcomes over all outcomes, when every outcome is equally likely.