Probability Academy: lottery maths step by step
Lotteries are one of the clearest examples of probability you can find: a fixed set of numbers, drawing without replacement and exact odds that anyone can check. This page gathers the calculators and articles on this site that explain the maths — for secondary-school students, first-year university students and anyone who wants to check the numbers themselves.
Learn by doing — four difficulty levels
Statistics, probability, mental math and map reading: 120 lessons each in four levels, with generated exercises and level tests.
Statistics tools
Probability tools
Where to start
1. Counting combinations
Why the order of numbers does not matter and how C(49, 6) = 13,983,816 is calculated.
2. Drawing without replacement
The hypergeometric distribution: N, K, n and k with urns, cards and lottery numbers.
3. Independent draws
Why hot and cold numbers do not predict the next draw, and what really changes with more tickets.
Calculators
Combinations, variations and permutations
Combinations, variations and permutations — nCr with every step shown.
Hypergeometric Distribution Calculator
P(X = k), P(X ≤ k), P(X ≥ k), the full table, mean and variance.
Lottery Odds Calculator
Exact odds of any number of matches, with one or two number pools.
Multiple Tickets Lottery Odds
The chance of at least one win over many tickets and draws: 1 − (1 − p)ⁿ.
Keno Odds Calculator
The full distribution of matches in Keno — a ready hypergeometric table.
Lottery System Calculator
How many bets a full system covers and what it costs.
Lottery Syndicate Calculator
Split a prize by member shares. For equal shares, enter 1 for each member.
Raffle Odds Calculator
Calculate the chance of at least one prize when winning tickets are drawn without replacement.
Scratch Card Expected Value Calculator
Enter the price, total ticket count and complete prize table for the same pool. Defaults are a worked example, not a real scratch card.
Articles
Guides
How Are Lottery Odds Calculated? Combinations, nCr and the 6/49 Example
Where do Lotto odds of 1 in 13,983,816 come from? We explain step by step what a combination is, why the order of numbers does not matter and how to calculate C…
read more… — How Are Lottery Odds Calculated? Combinations, nCr and the 6/49 Example
Guides
Hypergeometric Distribution Explained with Lottery, Cards and Urn Examples
Drawing without replacement in practice: what N, K, n and k mean, how to work out P(X = k) step by step and how the same formula describes balls in an urn, a ha…
read more… — Hypergeometric Distribution Explained with Lottery, Cards and Urn Examples
Statistics
What hot and cold numbers really mean
A hot number is one that has come up more often than the others over a chosen period. A cold one — less often. That's all it is: two words describing the past, …
Draw statistics
Historical draw statistics describe the past: how often each number came up. They are useful for checking that draws look random — not for predicting the next one. Lotto statistics.
All examples and exercises are written by us. They are practice material at secondary-school or introductory university level, not official exam tasks. No calculator here predicts draws or suggests numbers to play.