Conditional Probability and Bayes' Theorem Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Conditional Probability and Bayes' Theorem notes for GATE DA: 16 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Conditional Probability and Bayes

Probability · Unit 1 · Chapter 2

Conditional Probability & Bayes' Theorem

Reverse the direction of inference. Update beliefs with evidence. Classify with posteriors.

Core exam weapon

Stage 01 · Foundation

Conditional Probability & Multiplication Rule

Reshape the sample space when information arrives.

Stage 02 · Decomposition

Law of Total Probability

Break a hard event into a sum over cases.

Stage 03 · Reversal

Bayes' Theorem

Infer causes from observed effects.

Stage 04 · Application

Posterior Classification & Naive Bayes

From theorem to classifier, with misclassification risk.

Conditional Probability, Bayes and Posterior Classification

Topic 1 of 1 · Core

Conditional Probability & Bayes' Theorem

Condition on evidence. Reverse the inference. Classify with posteriors.

Conditioning intuition

Total probability

Bayes' theorem

Naive Bayes classifier

What Does It Mean to Condition?

Core idea. Learning that $B$ happened shrinks the sample space to $B$ . Every probability must be rescaled so that $B$ itself has probability one.

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) > 0

Numerator $P (A \cap B)$ : the part of $A$ that survives inside $B$ .
Denominator $P (B)$ : the rescaling factor so that $P (B ∣ B) = 1$ .

Equivalent form.

P (A \cap B) = P (A ∣ B) P (B)

. This is the multiplication rule, and it is often the first tool you reach for.

Multiplication Rule and Symmetry

Multiplication rule, two ways.

P (A \cap B) = P (A ∣ B) P (B) = P (B ∣ A) P (A)

Symmetry. Equating the two expressions gives

P (A ∣ B) P (B) = P (B ∣ A) P (A)

which is the seed of Bayes' theorem. Whenever a problem gives you $P (B ∣ A)$ and asks for $P (A ∣ B)$ , this symmetry is the starting point.

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Conditional Probability and Bayes' Theorem Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Conditional Probability and Bayes' Theorem notes for GATE DA: 16 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice quest

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