📐 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) ≥ 0

Maximum Value Rule

Probability of any event cannot exceed 1.

P(E) ≤ 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(∅) = 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 ∪ 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 ∪ B) = P(A) + P(B) − P(A ∩ B)

P(A or B) = P(A) + P(B) − P(A 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
EventCountProbability
King44 / 52
Heart1313 / 52
King of Hearts11 / 52
P(A ∪ B) = 4/52 + 13/52 − 1/52
P(A ∪ B) = 16/52 = 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 ∩ B) = P(A) × P(B)

Example

  • Tossing two coins:
P(HH) = 0.5 × 0.5 = 0.25

Case 2: Dependent Events

Occurrence of one event affects the other.

P(A ∩ B) = P(A) × P(B | A)

Where:

P(B | A) = Conditional Probability

Conditional Probability Rule

Probability of event B given A has occurred:

P(B | A) = {P(A ∩ B)}/{P(A)}

Example

From a deck of cards:

  • A = Card is a King
  • B = Card is a Heart
P(B | A) = 1 / 4

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

Summary Table of Probability Rules

RuleFormula
ComplementP(E′) = 1 − P(E)
Addition (Exclusive)P(A ∪ B) = P(A) + P(B)
Addition (Non-Exclusive)P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication (Independent)P(A ∩ B) = P(A)P(B)
ConditionalP(B | A) = P(A ∩ B) / P(A)
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