Tutorial

How to Calculate Probability: Formulas, Rules, and Examples

Published 14 March 2026 · 12 min read

Probability is the branch of mathematics that measures how likely an event is to occur. From weather forecasts to medical diagnoses, from card games to stock markets, probability underpins how we reason about uncertainty. Understanding the fundamentals of probability gives you a powerful framework for making better decisions.

In this guide, you will learn the core probability formulas, the rules for combining probabilities, and how to handle combinations and permutations. Each concept is illustrated with real-world examples you can follow step by step.

Key Concepts

Basic Probability Formula

The probability of an event is defined as:

Probability always falls between 0 (impossible) and 1 (certain). You can also express it as a percentage by multiplying by 100.

Complement Rule

The probability of an event not happening is:

This is useful when it is easier to calculate the probability of the opposite event.

Independent Events

Two events are independent if the outcome of one does not affect the other.

Conditional Probability

The probability of given that has occurred:

Combinations and Permutations

When counting outcomes, you often need to know how many ways you can choose or arrange items from a set.

Permutations (order matters)

Use when the arrangement of items matters (e.g. rankings, codes).

Combinations (order irrelevant)

Use when you are simply choosing a subset (e.g. lottery, teams).

Step-by-Step Worked Examples

Example 1: Rolling a Die

What is the probability of rolling a number greater than 4 on a fair six-sided die?

Favourable outcomes: 5, 6 (that is 2 outcomes). Total outcomes: 6.

Example 2: Drawing Cards

What is the probability of drawing a heart from a standard 52-card deck?

There are 13 hearts in a deck of 52 cards.

Example 3: Independent Events (Coin Flips)

What is the probability of flipping heads three times in a row with a fair coin?

Each flip is independent with .

That is a 12.5% chance.

Example 4: Conditional Probability

A bag contains 5 red marbles and 3 blue marbles. You draw one marble, keep it, then draw another. What is the probability that both marbles are red?

Probability the first is red: .

Given the first was red, there are now 4 red out of 7 total: .

Example 5: Combinations (Lottery)

A lottery requires you to choose 6 numbers from 49. How many possible combinations are there, and what is the probability of winning?

That is roughly a 1 in 14 million chance.

Example 6: The Addition Rule

What is the probability of drawing a king or a heart from a standard deck?

These events overlap (the king of hearts), so use the inclusion-exclusion principle:

Common Mistakes

Confusing independent and dependent events

Drawing cards without replacement makes events dependent. Drawing with replacement keeps them independent. Always check whether earlier outcomes change the remaining possibilities.

Adding probabilities for non-mutually-exclusive events

If events overlap, you must subtract the intersection. Otherwise you double-count outcomes that belong to both events.

Mixing up permutations and combinations

Ask yourself: does the order matter? Choosing a committee of 3 is a combination. Assigning president, vice president, and treasurer is a permutation.

The gambler's fallacy

Past results do not influence future independent events. A coin that has landed heads 10 times in a row still has a 50% chance of heads on the next flip.

Try It Yourself

Want to check your probability calculations? Our free probability calculator handles basic probability, combinations, permutations, and more. Enter your values and get instant results.

Open Probability Calculator

For counting arrangements and selections, try our Permutation & Combination Calculator.

15 Mar 20269 min read