Probability & Statistics

Syllabus Unit: Engineering Mathematics

Why it matters

Chapter overview

Probability and statistics quantify uncertainty — from the basic axioms and Bayes' theorem for updating beliefs given evidence, through named discrete/continuous distributions that model real engineering phenomena (defect counts, component lifetimes), to sampling theory and hypothesis testing that let engineers draw reliable conclusions from finite data — a core GATE Engineering Mathematics block.

General Aptitude & Mathematics Engineering Mathematics 15% weightage

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1Priors P(Ai)Known base rates

2Likelihoods P(B|Ai)Given evidence model

3Total probabilityP(B) = ΣP(B|Ai)P(Ai)

4Posterior P(Ai|B)Bayes' theorem output

1Probability Rules & Bayes' Theorem

Bayes' theorem is just conditional probability flipped around — it lets you update P(cause | evidence) once you know P(evidence | cause).

Learning Objectives

Apply the addition rule and the multiplication rule for independent/dependent events.
Compute conditional probability P(A|B) and check independence via P(A∩B)=P(A)P(B).
State and apply Bayes' theorem to compute a posterior probability given prior and likelihood.
Use the law of total probability to compute P(evidence) as a weighted sum over mutually exclusive causes.

Key Terms

Conditional ProbabilityP(A|B) = P(A∩B)/P(B), the probability of A given that B has already occurred.

Independent EventsEvents A and B where P(A∩B) = P(A)·P(B), so knowing one occurred gives no information about the other.

Bayes' TheoremA formula for reversing conditional probability: computing P(cause|evidence) from P(evidence|cause) and priors.

Law of Total ProbabilityP(B) = Σj P(B|Aj)·P(Aj), reconstructing an overall probability by summing over every mutually exclusive way it could occur.

The addition rule P(A∪B) = P(A) + P(B) − P(A∩B) and multiplication rule P(A∩B) = P(A)·P(B) (independent events) or P(A)·P(B|A) (dependent events) are the two building blocks every later probability question relies on.

Bayes' theorem

P(Ai|B) = [P(B|Ai)·P(Ai)] / Σj P(B|Aj)·P(Aj) — read the denominator as the law of total probability: it reconstructs P(B) by summing over every mutually exclusive way B could have occurred.

Classic pattern: several machines produce items with known defect rates and known production shares; given an item is found defective, Bayes' theorem finds the probability it came from a specific machine — reversing "cause→effect" (defect rate) into "effect→cause" (which machine).

Common MistakeThe most common GATE slip is plugging P(B|Ai) where P(Ai|B) is asked (or vice versa) — always double-check which probability is conditioned on which event before substituting.

Exam TipBuild a probability tree (prior branches, then likelihood branches) before writing Bayes' formula — it makes both the numerator and every term of the denominator visible at once.