Quantitative Aptitude · Probability

Basic Terms and Approaches to Probability for CA Foundation

Updated 1 October 2026 · Fact-checked

Probability measures how likely an event is, as a number from 0 to 1. First identify the random experiment and its sample space, then the event. Under the classical approach, P(A) = favourable outcomes ÷ total equally likely outcomes. Then check the event type and use the right definition.

Understand Basic Terms and Approaches to Probability

A random experiment is an activity with more than one possible result, where you cannot predict which one will occur in a single trial, even though you know all possible results. Tossing a coin, rolling a die and drawing a card are examples.

Each possible result is an outcome. The set of all outcomes is the sample space, written S. For one coin toss, S = {H, T}. For one die roll, S = {1, 2, 3, 4, 5, 6}. For two coin tosses, S = {HH, HT, TH, TT}, so it has 4 points.

An event is a subset of the sample space. "Getting an even number on a die" is the event {2, 4, 6}. A simple event has one outcome. A compound event has more than one.

Types of events you must know:

Mutually exclusive: cannot happen together in the same trial. A ∩ B is empty.
Exhaustive: together they cover the whole sample space. At least one must occur.
Equally likely: no outcome is favoured over another.
Independent: the occurrence of one does not change the probability of the other.
Complementary: A and A′ (not A). They are mutually exclusive and exhaustive.
Impossible event: the empty set, probability 0. Sure event: the whole sample space, probability 1.

There are three approaches to defining probability. The classical approach counts favourable outcomes when all outcomes are equally likely. The statistical (empirical) approach uses the long-run relative frequency from repeated trials. The axiomatic approach, due to Kolmogorov, does not say how to find probabilities. It lays down rules (axioms) that any probability assignment must satisfy.

Key formulas to remember

Classical probability: P(A) = m ÷ n; m = outcomes favourable to A, n = total outcomes in S. Valid only when outcomes are equally likely and exhaustive.
Statistical probability: P(A) = limit of (f ÷ n) as n → ∞; f = number of times A occurs in n trials. In practice you use a large n as an estimate.
Axiom 1 (non-negativity): 0 ≤ P(A) ≤ 1; Holds for every event A.
Axiom 2 (certainty): P(S) = 1; The sure event has probability 1.
Axiom 3 (additivity): P(A ∪ B) = P(A) + P(B); Only for mutually exclusive events A and B.
Complement rule: P(A′) = 1 − P(A); Useful for 'at least one' questions.
Odds: Odds in favour of A = m : (n − m); Then P(A) = m ÷ n. Odds against = (n − m) : m.
Exhaustive and exclusive set: P(A₁) + P(A₂) + ... + P(Aₖ) = 1; For events that are mutually exclusive and exhaustive.

How to solve Basic Terms and Approaches to Probability questions

Use this order for any question on basic terms or the approaches to probability.

1Read the experiment and write down what one trial is (one toss, one draw, two dice together).
2Find the size of the sample space n. For two dice n = 36, for three coins n = 8, for a deck n = 52.
3Write the event in words, then list or count its favourable outcomes m.
4Check the outcomes are equally likely. If not, the classical formula does not apply; use given frequencies or axioms.
5Compute P(A) = m ÷ n, or use 1 − P(A′) if counting the complement is easier.
6For questions on event types, test the definitions: do the events overlap (exclusive?), do they cover S (exhaustive?).
7Check that your answer lies between 0 and 1, and that probabilities of exclusive and exhaustive events add to 1.

Quickest way: Count the complement and test the options

When to use it: Use for MCQs where the event is 'at least' or 'not', or where the options include values above 1 or a definition match.

Reject any option that is negative or greater than 1 at once.
Write n first (6, 36, 8, 52). The answer's denominator should divide into it.
If the event says 'at least one', compute 1 − P(none).
For definition questions, recall: exclusive means no common outcome; exhaustive means nothing is left out; both together means a partition of S.
Skip questions with long sample-space listings if you are short of time; wrong answers cost 0.25 marks.

Common mistakes in Basic Terms and Approaches to Probability

Confusing mutually exclusive with independent.
Both sound like 'unrelated' events.
Fix: Exclusive events cannot occur together. If both have non-zero probability, they are in fact dependent, because one occurring rules out the other.
Writing the sample space of two dice as 12 outcomes.
Adding 6 + 6 instead of multiplying.
Fix: Each die has 6 outcomes and they combine, so n = 6 × 6 = 36.
Using the classical formula when outcomes are not equally likely.
Students count outcomes blindly, such as 'rain or no rain = 1/2'.
Fix: Ask whether each outcome has the same chance. If not, use relative frequency or the given probabilities.
Thinking mutually exclusive events are always exhaustive.
Two events like 'head' and 'tail' are both, so students generalise.
Fix: On a die, {1} and {2} are exclusive but not exhaustive. Exhaustive needs the union to equal S.
Treating the statistical probability as exact after a few trials.
Seeing 6 heads in 10 tosses and calling P(H) = 0.6.
Fix: The relative frequency settles near the true value only for a large number of trials.
Giving odds as the probability.
Odds 2 : 3 looks like 2/3.
Fix: Odds in favour 2 : 3 means m = 2, n = 5, so P = 2/5.

Worked examples

Example 1

Two fair dice are thrown together. The probability that the sum of the numbers is 9 is: (a) 1/12 (b) 1/9 (c) 1/6 (d) 5/36

Show the solution

Sample space size n = 6 × 6 = 36.
Favourable outcomes with sum 9: (3,6), (4,5), (5,4), (6,3). So m = 4.
P = 4 ÷ 36 = 1/9.

Answer: (b) 1/9

Example 2

A bag has 3 red and 2 blue balls. One ball is drawn at random. The odds in favour of drawing a blue ball are: (a) 2 : 3 (b) 3 : 2 (c) 2 : 5 (d) 3 : 5

Show the solution

Total outcomes n = 5, equally likely.
Favourable (blue) m = 2.
Unfavourable = 5 − 2 = 3.
Odds in favour = m : (n − m) = 2 : 3.

Answer: (a) 2 : 3

Example 3

A coin is tossed three times. The probability of getting at least one head is: (a) 1/8 (b) 3/8 (c) 7/8 (d) 1/2

Show the solution

Sample space size n = 2 × 2 × 2 = 8.
Complement of 'at least one head' is 'no head', that is TTT only. So P(no head) = 1/8.
P(at least one head) = 1 − 1/8 = 7/8.

Answer: (c) 7/8

Exam tips

Questions on this topic are often definition-based. Learn the exact meaning of mutually exclusive, exhaustive, equally likely and independent.
Memorise standard sample space sizes: coin 2, die 6, two dice 36, three coins 8, deck 52.
For 'at least one', go straight to the complement.
If an option is above 1 or below 0, eliminate it immediately.
Know which approach fits: equally likely outcomes means classical, repeated data means statistical, rules for probability means axiomatic.

Practice questions from Probability

Basic Terms and Approaches to Probability: frequently asked questions

What is the difference between the classical and statistical approach to probability?

The classical approach counts favourable outcomes out of equally likely outcomes, without doing any experiment. The statistical approach uses the relative frequency from many actual trials. Use classical for fair coins and dice, and statistical when you have observed data.

What are the axioms of probability?

Every probability must be between 0 and 1, the probability of the whole sample space is 1, and for mutually exclusive events the probability of the union is the sum of their probabilities. Everything else in probability is built from these rules.

Can two events be both mutually exclusive and exhaustive?

Yes. Such events split the sample space with no overlap, so their probabilities add to 1. An event and its complement are the simplest example.

Are mutually exclusive events independent?

No, not when both have non-zero probability. If one occurs, the other cannot, so the occurrence of one changes the chance of the other to zero.