🔁 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:

$P(B \mid A)$

But many real problems ask:

$P(A \mid B)$

Bayes' theorem connects both.

Bayes' Theorem Formula

Mathematical Formula

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}$

Where:

• $P(A)$ → Prior probability
• $P(B \mid A)$ → Likelihood
• $P(B)$ → Evidence
• $P(A \mid B)$ → Posterior probability

Probability Terminology

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:

$P(+ \mid D) = 0.99$

• Test positive if disease NOT present:

$P(+ \mid \overline{D}) = 0.05$

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

Step 1: Define Events

• D → Person has disease
• $\overline{D}$ → Person does not have disease
• + → Test is positive

Step 2: Write Given Data

Step 3: Calculate Evidence P(+)

$P(+) = P(+ \mid D) \times P(D) + P(+ \mid \overline{D}) \times P(\overline{D})$

Step 4: Apply Bayes' Theorem

$P(D \mid +) = \frac{P(+ \mid D) \times P(D)}{P(+)}$

$P(D \mid +) = \frac{0.0099}{0.0594} \approx 0.1667$

$P(D \mid +) \approx 0.1667$

Step 5: Final Answer

$\text{Probability that a person actually has disease after testing positive} \approx 16.67\%$

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

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