🔁 Bayes’ Theorem

Last Updated: Jan 2026

Bayes’ Theorem helps us update probability when new information is available.

It answers questions like:

  • What is the probability of cause, given the result?
  • How to reverse conditional probability

Bayes’ theorem is the backbone of:

  • Machine Learning (Naive Bayes)
  • Medical diagnosis
  • Spam filtering
  • Decision making

🗣 Hinglish Tip: Bayes = result dekh kar cause ka chance nikalna

Why Bayes’ Theorem is Needed?

Conditional probability gives:

P(B | A)

But many real problems ask:

P(A | B)

Bayes’ theorem connects both.

Bayes’ Theorem Formula

Mathematical Formula

P(A | B) = P(B | A) × P(A) / P(B)

Where:

  • P(A) → Prior probability
  • P(B | A) → Likelihood
  • P(B) → Evidence
  • P(A | B) → Posterior probability

Probability Terminology

TermMeaning
PriorInitial belief (before evidence)
LikelihoodProbability of evidence given cause
PosteriorUpdated probability (after evidence)
EvidenceTotal probability of evidence

Example (Medical Test )

A disease affects 1% of population.

  • Probability that a person has disease:
P(D) = 0.01

Test accuracy:

  • Test positive if disease present:
P(+ | D) = 0.99
  • Test positive if disease NOT present:
P(+ | D̄) = 0.05

👉 If a person tests positive, find probability that the person actually has disease.

Step 1: Define Events

  • D → Person has disease
  • D̄ → Person does not have disease
  • + → Test is positive

Step 2: Write Given Data

QuantityValue
P(D)0.01
P(D̄)0.99
P(+ | D)0.99
P(+ | D̄)0.05

Step 3: Calculate Evidence P(+)

P(+) = P(+ | D) × P(D) + P(+ | D̄) × P(D̄)
TermCalculationValue
P(+ | D) × P(D)0.99 × 0.010.0099
P(+ | D̄) × P(D̄)0.05 × 0.990.0495
Total P(+)—0.0594

Step 4: Apply Bayes’ Theorem

P(D | +) = P(+ | D) × P(D) / P(+)

P(D | +) = 0.0099 / 0.0594

P(D | +) ≈ 0.1667

Step 5: Final Answer

Probability that a person actually has disease after testing positive ≈ 16.67%

🗣 Hinglish Tip: Test positive hone ka matlab confirm disease nahi hota

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