Fundamentals of Business Mathematics and Statistics · Probability

Basic Concepts of Probability for CMA Foundation

Updated 10 October 2026 · Fact-checked

Probability measures how likely an event is, as a number from 0 to 1. In the classical definition, P(A) = favourable outcomes ÷ total equally likely outcomes. To solve a question, list the sample space, count the outcomes in the event, and divide. Check whether events are mutually exclusive, exhaustive or independent.

Understand Basic Concepts of Probability

A random experiment is an action whose result you cannot predict with certainty, though you know all possible results. Tossing a coin, rolling a die and drawing a card are examples. Each single result is an outcome.

The sample space (S) is the set of all possible outcomes. For one die, S = {1, 2, 3, 4, 5, 6}. An event is any subset of the sample space, such as 'getting an even number' = {2, 4, 6}. An event with one outcome is a simple event. An event with more than one outcome is a compound event.

Events are classified by how they relate to each other:

Mutually exclusive: they cannot happen together. On one die, 'even' and 'odd' are mutually exclusive.
Exhaustive: together they cover the whole sample space, so at least one must happen.
Equally likely: no outcome is favoured over another.
Independent: the occurrence of one does not change the probability of the other. Two coin tosses are independent.

Mutually exclusive and independent are different ideas. If two events with non-zero probability are mutually exclusive, they cannot be independent. Knowing one happened tells you the other did not.

There are three definitions of probability. The classical definition uses equally likely outcomes and needs no experiment. The statistical (empirical) definition uses relative frequency: the fraction of times the event occurs in a very large number of trials. The axiomatic definition sets rules: P(A) ≥ 0, P(S) = 1, and for mutually exclusive events P(A ∪ B) = P(A) + P(B). Any valid probability lies between 0 and 1.

Key formulas to remember

Classical probability: P(A) = m ÷ n; m = outcomes favourable to A, n = total outcomes. Valid only when all outcomes are equally likely.
Range of probability: 0 ≤ P(A) ≤ 1; P = 0 for an impossible event and P = 1 for a certain event.
Complement rule: P(A') = 1 − P(A); Use it for 'at least one' or 'not' questions.
Odds in favour and against: Odds in favour = m : (n − m); odds against = (n − m) : m; Convert odds a : b in favour to probability a ÷ (a + b).
Statistical probability: P(A) = limit of (f ÷ N) as N becomes very large; f = times A occurred, N = number of trials.
Mutually exclusive events: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B); Applies only when A and B cannot occur together.
Independent events: P(A ∩ B) = P(A) × P(B); This is the test for independence.
Exhaustive and mutually exclusive set: P(E1) + P(E2) + … + P(Ek) = 1; Holds for events that are both exhaustive and mutually exclusive.

How to solve Basic Concepts of Probability questions

Use this method for any basic probability question in the paper.

1Identify the random experiment and write down what is being done (one die, two coins, a card, a bag of balls).
2Find the total number of equally likely outcomes, n. For two coins n = 4, for two dice n = 36, for a pack of cards n = 52.
3Define the event clearly and count the favourable outcomes, m.
4Check the equally likely condition. If outcomes are not equally likely, use the given relative frequencies instead.
5Compute P(A) = m ÷ n. If the event is 'at least one' or 'not', consider 1 − P(complement).
6For a question on types of events, test each pair: can they occur together (mutually exclusive)? Do they cover all outcomes (exhaustive)? Does P(A ∩ B) = P(A) × P(B) (independent)?
7Check the answer lies between 0 and 1, then pick the matching option.

Quickest way: Count and complement

When to use it: Use this for most MCQs on coins, dice, cards and balls, where counting is quick.

Memorise the standard sizes: coin 2, die 6, two dice 36, cards 52 (13 per suit, 4 suits, 12 face cards, 26 red).
For two dice, list only the favourable pairs and divide by 36.
If the question says 'at least one', compute 1 − P(none) instead of counting many cases.
For odds a : b in favour, the probability is a ÷ (a + b). Do not divide by b.
Eliminate any option above 1 or negative. For type-of-event questions, eliminate options that call two events with non-zero probability both mutually exclusive and independent.

Common mistakes in Basic Concepts of Probability

Treating mutually exclusive and independent as the same thing.
Both seem to mean 'unrelated', so students mix them up.
Fix: Mutually exclusive means cannot occur together. Independent means one does not affect the other's probability. For non-zero probabilities, mutually exclusive events are dependent.
Using m ÷ n when outcomes are not equally likely.
Students count outcomes without checking fairness, for example treating 'sum of two dice' values 2 to 12 as equally likely.
Fix: Always list the underlying equally likely outcomes, such as the 36 pairs for two dice.
Writing odds as probability.
Odds 3 : 2 in favour is read as 3/2 or 3/5 loosely.
Fix: Odds in favour a : b means probability a ÷ (a + b). Here it is 3/5.
Counting two dice outcomes as 6 + 6 = 12 or listing (1,2) and (2,1) as the same.
Students forget that the dice are distinct.
Fix: Total outcomes are 6 × 6 = 36, and (1,2) and (2,1) are different outcomes.
Giving a probability greater than 1 or a negative value.
Students divide total by favourable by mistake.
Fix: Keep favourable outcomes in the numerator. Any answer outside 0 to 1 is wrong.
Confusing classical and statistical definitions.
Both give a ratio, so the difference is missed.
Fix: Classical uses equally likely outcomes with no trials. Statistical uses observed relative frequency over a large number of trials.

Worked examples

Example 1

Two fair dice are thrown together. What is the probability that the sum of the numbers is 8?

Show the solution

Total outcomes n = 6 × 6 = 36.
Pairs with sum 8: (2,6), (3,5), (4,4), (5,3), (6,2). So m = 5.
P = m ÷ n = 5 ÷ 36.

Answer: 5/36

Example 2

The odds in favour of an event are 3 : 5. What is the probability that the event does not occur?

Show the solution

Odds in favour 3 : 5 means P(event) = 3 ÷ (3 + 5) = 3/8.
Use the complement rule: P(not event) = 1 − 3/8.
1 − 3/8 = 5/8.

Answer: 5/8

Exam tips

Questions on types of events are usually conceptual. Learn the four definitions word for word in your own language, and practise telling mutually exclusive from independent.
Many MCQs hide a complement. When you see 'at least one', calculate 1 − P(none) first.
Know the pack of 52 cards cold, including 4 kings, 12 face cards and 26 red cards, so you do not waste time counting.
There is no negative marking, so if time is short, eliminate options above 1 and guess among the rest.

Practice questions from Probability

Basic Concepts of Probability in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationBasic Terms and Approaches to Probability

Basic Concepts of Probability: frequently asked questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. Independent events do not affect each other, so P(A ∩ B) = P(A) × P(B). If both events have non-zero probability, they cannot be both.

What is the difference between classical and statistical probability?

Classical probability is worked out in advance as favourable ÷ total outcomes, assuming equally likely outcomes. Statistical probability is found from experiments, as the relative frequency of the event over a very large number of trials.

What is a sample space with an example?

The sample space is the set of all possible outcomes of a random experiment. For tossing two coins, S = {HH, HT, TH, TT}, which has 4 outcomes.

Can probability be negative or more than 1?

No. Probability always lies from 0 to 1. A value of 0 means the event is impossible and 1 means it is certain.