Probability & Basic Statistics

Syllabus Unit: Mathematics

Why it matters

Chapter overview

Probability and basic statistics cover the classical definition of probability and its theorems, combinatorial (counting-based) probability problems, and the three measures of central tendency (mean, median, mode) alongside variance and standard deviation as measures of spread — a compact unit that rewards careful formula application over memorisation.

Mathematics & General Ability Mathematics 40% weightage

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Event A

1Sample Space (S)The complete set of possible outcomes

2Complement (1−P(A))Probability that A does NOT occur

3Union with BP(A∪B) = P(A)+P(B)−P(A∩B)

4Intersection with BP(A∩B), both events occurring together

1Probability Basics & Theorems

Probability quantifies uncertainty on a scale from 0 (impossible) to 1 (certain), built from the ratio of favourable to total equally likely outcomes.

Learning Objectives

Define probability as favourable outcomes / total outcomes for equally likely events.
Apply the addition theorem P(A∪B) = P(A)+P(B)−P(A∩B).
Distinguish independent, mutually exclusive and exhaustive events.
Apply conditional probability P(A|B) = P(A∩B)/P(B) and the idea behind Bayes' theorem.

Key Terms

Sample Space (S)The set of all possible outcomes of a random experiment.

Mutually Exclusive EventsEvents that cannot occur together, so P(A∩B) = 0.

Independent EventsEvents where one's occurrence doesn't affect the other's probability, P(A∩B) = P(A)·P(B).

Conditional ProbabilityP(A|B), the probability of A occurring given that B has already occurred.

Probability measures how likely an event is, ranging from 0 (impossible) to 1 (certain), calculated for equally likely outcomes as a simple ratio.

Classical definition

P(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1

Addition theorem

P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events, P(A∪B)=P(A)+P(B)

Common MistakeForgetting to subtract P(A∩B) is the single most common NDA probability error.

Independent vs mutually exclusive

Independent events can both occur (e.g. two separate coin tosses); mutually exclusive events by definition cannot both occur (e.g. drawing a red OR a black ball in one draw) — these are opposite ideas, not the same thing.

Conditional probability

P(A|B) = P(A∩B) / P(B)

Bayes' theorem reverses this relationship — it updates the probability of a cause given an observed effect, combining conditional probabilities in both directions.