🎲 Probability

Last Updated: Jan 2026

Probability is the branch of mathematics that deals with measuring uncertainty. It tells us how likely an event is to occur.

Probability values always lie between:

0 ≤ P(Event) ≤ 1

It uses:

  • Foundation of Statistics
  • Core of Machine Learning
  • Used in:
    • Risk analysis
    • Games & simulations
    • Decision making under uncertainty

🗣 Hinglish Tip: Probability = chance ka mathematical measurement

There are Few main things in probability to understand:

  • Random experiment
  • Outcome
  • Sample space
  • Event

Random Experiment

A random experiment is an experiment whose outcome cannot be predicted with certainty.

Examples

  • Tossing a coin
  • Rolling a dice
  • Drawing a card

Outcome

An outcome is a single possible result of a random experiment.

Examples

  • Coin toss → Head (H) or Tail (T)
  • Dice roll → 1, 2, 3, 4, 5, 6

Notation:

Outcome → ω

Sample Space

The sample space is the set of all possible outcomes of a random experiment.

Notation:

Sample Space → S

Examples

  • Coin toss:
S = {H, T}
  • Dice roll:
S = {1, 2, 3, 4, 5, 6}

Event

An event is a subset of the sample space.

Notation:

Event → E

Example

  • Event: getting an even number on dice
E = {2, 4, 6}

🗣 Hinglish Tip: Event = sample space ka chosen part

Types of Events

Simple Event

An event with only one outcome.

Example:

E = {3}

Mutually Exclusive Events

Two events that cannot occur together.

Example:

  • Head and Tail in one coin toss

Independent Events

Events that do not affect each other.

Example:

If a coin is tossed twice, the outcomes are independent.

Dependent Events

Events that affect each other. Example:

  • If we draw a card, the rank of the card depends on the suit.

Complementary Events

Events that complement each other.

Example:

  • Head and Tail in one coin toss
E = {H, T}

Not H = 1- H

Types of Probability

Classical Probability

Based on mathematical formula. Used when outcomes are equally likely.

Formula:

P(E) = favourable outcomes / total trials

Example:

  • Probability of getting even number on dice:
P(E) = 3 / 6 = 1 / 2

Empirical (Experimental) Probability

Based on actual experiments.

Formula:

P(E)=(Numberoftimeseventoccurs)/(Totaltrials)

Subjective Probability

Based on personal judgment or experience.

Example:

  • Chance of rain tomorrow
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