Build probability from the ground up. Understand what a probability actually measures, how to define sample spaces and events rigorously, and master the three Kolmogorov axioms that underpin all of modern probability theory.
12 Worked Examples10-Question Quiz1 500+ WordsKolmogorov AxiomsCounting Rules
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§ 01What Is Probability? The Core Idea
Probability is the mathematics of uncertainty. It gives us a precise, consistent language for describing how likely events are — from rolling a die to modelling financial risk to building machine-learning classifiers.
At its heart, a probability is a number between 0 and 1 assigned to an event. A probability of 0 means the event is impossible; a probability of 1 means it is certain. Every other value sits somewhere on this scale — a probability of 0.5 means the event is equally likely to occur or not occur.
Before we can calculate any probabilities, we need to define the mathematical structure we are working in. That structure has three components: the sample space, events, and a probability function that obeys Kolmogorov’s axioms. Get these three things right and everything else follows logically.
The Probability Scale
0 ≤ P(A) ≤ 1 for any event A
P(impossible) = 0 P(certain) = 1
§ 02Sample Spaces and Events
A sample space collects every possible outcome of an experiment into one set. An event is any subset of that sample space — any collection of outcomes we care about.
The Sample Space Ω
The sample space Ω (capital omega) is the set of all possible outcomes of a random experiment. Every outcome must be:
Mutually exclusive — no two outcomes can happen at the same time in a single trial. Collectively exhaustive — at least one outcome from Ω must occur every time the experiment runs.
Experiment
Sample Space Ω
Size |Ω|
Flip one coin
{H, T}
2
Roll one fair die
{1, 2, 3, 4, 5, 6}
6
Flip two coins
{HH, HT, TH, TT}
4
Roll two dice
{(1,1),(1,2), …, (6,6)}
36
Pick a card from a standard deck
All 52 cards
52
Time until next bus (minutes)
[0, ∞)
Uncountably infinite
Events
An event A is any subset of Ω. It can contain one outcome (a simple event), several outcomes (a compound event), all outcomes (the certain event Ω), or no outcomes (the impossible event ∅).
Example 01 — Listing Events
A fair die is rolled once. Write out the events: A = “even number”, B = “greater than 4”, C = “even AND greater than 4”
Ω
Ω = {1, 2, 3, 4, 5, 6}
Event A
A = {2, 4, 6} — all even numbers in Ω
Event B
B = {5, 6} — all numbers greater than 4
Event C
C = A ∩ B = {6} — even AND greater than 4: only 6
A = {2,4,6} B = {5,6} C = A∩B = {6}
§ 03Kolmogorov’s Axioms — The Three Rules All Probabilities Must Obey
In 1933 Andrey Kolmogorov published three axioms that define what a probability function must be. Every result in probability theory — from basic rules to the Central Limit Theorem — follows from these three statements alone.
✦ Kolmogorov’s Three Axioms of Probability
Let Ω be a sample space and P a function that assigns a number to each event. P is a valid probability function if and only if:
Axiom 1 (Non-negativity): P(A) ≥ 0 for every event A. Axiom 2 (Normalisation): P(Ω) = 1 (something must happen). Axiom 3 (Countable Additivity): If A and B are mutually exclusive (A∩B = ∅), then P(A∪B) = P(A) + P(B).
Every other rule in probability is a theorem derived from these three axioms. Let’s derive the most important ones immediately.
Derived Rule 1 — The Complement Rule
The complement A′ (also written Aᶜ or Â) is the event “A does not occur” — i.e. all outcomes in Ω that are not in A. Since A and A′ are mutually exclusive and together cover all of Ω:
Complement Rule
P(A′) = 1 − P(A)
Derived Rule 2 — The Addition Rule
When A and B are not mutually exclusive, their union double-counts the outcomes in A∩B. We subtract the overlap once:
General Addition Rule
P(A∪B) = P(A) + P(B) − P(A∩B)
When A and B are mutually exclusive, P(A∩B) = 0 and the rule reduces to Axiom 3.
§ 04Equally Likely Outcomes — The Classical Formula
When every outcome in a finite sample space is equally likely, probability reduces to counting. This is the classical model and the starting point for most introductory problems.
Classical Probability Formula
P(A) = |A| / |Ω| = (number of favourable outcomes) / (total number of outcomes)
This formula only applies when every outcome is equally likely — a fair coin, a fair die, a well-shuffled deck. It does not apply to a biased coin or to outcomes on a continuous scale.
Example 02 — Single Die
A fair six-sided die is rolled. Find P(rolling a 3), P(rolling an even number), P(rolling a number greater than 4)
The probability that it rains tomorrow is 0.35. What is the probability that it does NOT rain?
Apply
P(no rain) = 1 − P(rain) = 1 − 0.35 = 0.65
P(no rain) = 0.65
§ 05Counting Rules — Permutations, Combinations & the Multiplication Principle
Many probability problems reduce to counting favourable outcomes and total outcomes. These three counting tools cover the vast majority of cases you will encounter.
The Multiplication Principle
If a task consists of k sequential steps, with n1 ways to do step 1, n2 ways to do step 2, …, nk ways to do step k, then the total number of ways to complete the entire task is n1 × n2 × … × nk.
Permutations — Order Matters
Permutations — Ordered Arrangements of r Items from n
P(n, r) = n! / (n−r)!
Combinations — Order Does Not Matter
Combinations — Unordered Selections of r Items from n
C(n, r) = nCr = n! / [r!(n−r)!]
Example 06 — Multiplication Principle
A restaurant menu has 4 starters, 6 mains, and 3 desserts. How many different three-course meals are possible?
Apply
Total = 4 × 6 × 3 = 72
72 different three-course meals
Example 07 — Permutations
In how many ways can 3 books be arranged on a shelf chosen from 8 different books?
Identify
Order matters (which book is first, second, third). Use P(8,3).
Calculate
P(8,3) = 8! / (8−3)! = 8! / 5! = 8×7×6 = 336
336 arrangements
Example 08 — Combinations & Probability
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Find P(both red)
Total ways
Choose 2 from 8: C(8,2) = 8!/(2!6!) = 28
Favourable
Choose 2 red from 5: C(5,2) = 5!/(2!3!) = 10
P(both red)
P = 10/28 = 5/14
P(both red) = 5/14 ≈ 0.357
Example 09 — Lottery Probability
A lottery requires choosing 6 numbers from 1 to 49. What is the probability of matching all 6?
Total outcomes
C(49,6) = 49!/(6!43!) = 13,983,816
Favourable
Only 1 winning combination
P(jackpot)
1 / 13,983,816 ≈ 0.0000000715
P(jackpot) ≈ 1 in 13.98 million
§ 06More Rules — Addition, Mutually Exclusive & Probability Tables
Example 10 — Addition Rule (Non-Exclusive Events)
P(A) = 0.4, P(B) = 0.5, P(A∩B) = 0.2. Find P(A∪B)
Formula
P(A∪B) = P(A) + P(B) − P(A∩B)
Substitute
= 0.4 + 0.5 − 0.2 = 0.7
P(A∪B) = 0.7
Example 11 — Mutually Exclusive Events
A single card is drawn from a deck. A = “King”, B = “Queen”. Find P(A∪B)
Check
A card cannot be both a King and a Queen. So A∩B = ∅ — mutually exclusive.
Apply
P(A∪B) = P(A) + P(B) = 4/52 + 4/52 = 8/52 = 2/13
P(King or Queen) = 2/13
Example 12 — Using All the Rules Together
In a class of 30 students, 18 study Maths, 15 study Physics, and 10 study both. A student is chosen at random. Find: P(Maths only), P(Physics only), P(neither)
P(M∪P)
18/30 + 15/30 − 10/30 = 23/30
P(Maths only)
In Maths but not Physics: 18 − 10 = 8 students. P = 8/30 = 4/15
P(Physics only)
15 − 10 = 5 students. P = 5/30 = 1/6
P(neither)
1 − 23/30 = 7/30
Maths only: 4/15 Physics only: 1/6 Neither: 7/30
§ 07Continue in Probability & Statistics
Basic probability is the foundation. Every topic below builds directly on what you’ve learned here — from conditioning on new information to modelling real-world data with distributions.
✦ All Probability & Statistics Topics
Select any topic to continue your study — you are currently on Basic Probability.
⚠ Mistake 1 — Confusing P(A∪B) with P(A) + P(B)
Adding probabilities directly only works when A and B are mutually exclusive. If they can overlap, you must subtract P(A∩B). Forgetting this leads to probabilities greater than 1 — an immediate red flag.
⚠ Mistake 2 — Using the Classical Formula When Outcomes Are Not Equally Likely
The formula P(A) = |A|/|Ω| only applies when every single outcome has the same probability. If a die is biased or a bag contains different-sized balls, you cannot count outcomes and divide.
⚠ Mistake 3 — Mixing Up Permutations and Combinations
Permutations count ordered arrangements; combinations count unordered selections. Ask yourself: “Does the order in which items are chosen matter?” Choosing a 4-digit PIN uses permutations; choosing a committee uses combinations.
⚠ Mistake 4 — P(A∩B) = P(A) × P(B) Only When Independent
The product rule P(A∩B) = P(A) × P(B) holds only for independent events. When events are dependent (e.g. drawing cards without replacement), you must use the general rule P(A∩B) = P(A) × P(B|A), covered in Conditional Probability.
✦ ✦ ✦
§ 09Test Your Knowledge — 10-Question Quiz
Select your answer for each question then click Submit All Answers. Correct answers light up green; wrong answers reveal the right choice with a full explanation.
Score: 0 / 0
Question 1 of 10
A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?
Question 2 of 10
According to Kolmogorov’s axioms, which of the following must always be true?
Question 3 of 10
P(A) = 0.6. What is P(A′)?
Question 4 of 10
P(A) = 0.3, P(B) = 0.5, P(A∩B) = 0.1. What is P(A∪B)?
Question 5 of 10
Two fair coins are tossed. What is the probability of getting at least one tail?
Question 6 of 10
How many ways can 4 people be seated in a row of 4 chairs?
Question 7 of 10
A committee of 3 is chosen from 7 people. How many ways can this be done?
Question 8 of 10
A card is drawn from a 52-card deck. What is P(King or Spade)?
Question 9 of 10
Events A and B are mutually exclusive. P(A) = 0.25, P(B) = 0.4. What is P(A∪B)?
Question 10 of 10
A bag has 4 red and 6 blue balls. One ball is drawn at random. What is the probability it is NOT red?