Conditional Probability

Last Updated: Jan 2026

Conditional Probability measures the probability of an event when another event has already occurred.

It answers questions like:

• What is the chance of B, given A has happened?
• How probability changes when conditions are applied

This concept is core to:

• Bayes' Theorem
• Machine Learning (Naive Bayes)
• Statistics & Data Science

🗣 Hinglish Tip: Conditional Probability = condition lagne ke baad probability nikalna

Basic Notation

• A, B → Events
• $P(A)$ → Probability of event A
• $P(B \mid A)$ → Probability of B given A has occurred

The symbol | means “given that”

Conditional Probability Formula

Mathematical Definition

$P(B \mid A) = \frac{P(A \cap B)}{P(A)}, \quad P(A) \neq 0$

Meaning

• Numerator → Probability of both A and B
• Denominator → Probability of A
• Sample space becomes restricted to A

Conceptual Understanding (Very Important)

When condition A happens:

• We discard outcomes where A did not occur
• Probability is recalculated within event A only

🗣 Hinglish Tip: Condition lagte hi sample space chhota ho jaata hai

Example 1: Cards (Classic College Example)

A card is drawn from a standard deck of 52 cards.

• Event A: Card is a King
• Event B: Card is a Heart

Step 1: Find probabilities

Step 2: Apply Formula

$P(B \mid A) = \frac{1}{52}/ \frac{4}{52} = \frac{1}{4}$

Probability that the card is a Heart given it is a King = 1/4

Example 2: Dice (Independent vs Conditional)

A fair die is rolled.

• A: Number is even
• B: Number is greater than 3

Step 1: Identify sample spaces

Step 2: Calculate

$P(A) = \frac{3}{6}
\\[0.5em]
P(A \cap B) = \frac{2}{6}$

$P(B \mid A) = {\frac{2}{6}}/{\frac{3}{6}} = \frac{2}{3}$

Conditional Probability Formula (Rearranged)

From definition:

$P(A \cap B) = P(A) \times P(B \mid A)$

This is used heavily in:

• Joint probability
• Bayes' theorem
• ML models

Conditional Probability vs Independent Events

Independent Events Rule

If A and B are independent:

$P(B \mid A) = P(B)$

Meaning:

• Event A does not affect event B

Example

Coin tosses:

$P(\text{Head on 2nd toss} \mid \text{Head on 1st toss}) = 0.5$

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