📐 Probability Rules

Last Updated: Jan 2026

Probability rules are fundamental laws that help us calculate probabilities correctly for different types of events.

These rules are the building blocks for:

• Conditional Probability
• Bayes' Theorem
• Random Variables
• Machine Learning models

🗣 Hinglish Tip: Probability rules = chance calculate karne ke fixed kanoon

Basic Probability Rules

Non-Negativity Rule

Probability of any event cannot be negative.

$P(E) \geq 0$

Maximum Value Rule

Probability of any event cannot exceed 1.

$P(E) \leq 1$

Certain Event Rule

If an event is sure to occur:

$P(S) = 1$

Where S is the sample space.

Impossible Event Rule

If an event cannot occur:

$P(\emptyset) = 0$

Complement Rule

If E is an event, then its complement E' is:

$P(E') = 1 - P(E)$

Example

If:

$P(Rain) = 0.3$

Then:

$P(No Rain) = 1 - 0.3 = 0.7$

🗣 Hinglish Tip: Complement = jo event nahi hua

Addition Rule of Probability

Used when finding probability of A or B.

Case 1: Mutually Exclusive Events

Events that cannot occur together.

$P(A \cup B) = P(A) + P(B)$

Example

• Probability of Head: P(H) = 0.5
• Probability of Tail: P(T) = 0.5

$P(H or T) = 0.5 + 0.5 = 1$

Case 2: Non-Mutually Exclusive Events

Events that can occur together.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)

\\[0.5em]

P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B)$

Where:

• A and B are events
• P(A and B) means the probability of both events A and B occurring together
• P(A or B) means the probability of either event A or B occurring

Example

A card is drawn from a deck.

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

$P(A \cup B) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52}

\\[0.5em]

P(A \cup B) = \frac{16}{52} = \frac{4}{13}$

Multiplication Rule of Probability

Used when finding probability of A and B.

Case 1: Independent Events

Occurrence of one event does not affect the other.

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

Example

• Tossing two coins:

$P(HH) = 0.5 \times 0.5 = 0.25$

Case 2: Dependent Events

Occurrence of one event affects the other.

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

Where:

$P(B \mid A) = \text{Conditional Probability}$

Conditional Probability Rule

Probability of event B given A has occurred:

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

Example

From a deck of cards:

• A = Card is a King
• B = Card is a Heart

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

🗣 Hinglish Tip: Condition lag jaaye → sample space chhota ho jaata hai

Summary Table of Probability Rules

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