Lottery advertising rarely mentions the odds, and when it does the numbers are so large that they are hard to picture. Working out a few of them yourself is the best way to see how lotteries are built. This quiz picks a random game format – 6 of 49, 6 of 59, 5 of 69 and others – and asks a question about it, with the arithmetic you need explained in the solution.
There are five kinds of question: the odds of the jackpot, the odds of matching exactly some of your numbers, the odds of a jackpot that also needs a bonus ball from a separate drum, whether matching more numbers can be more likely than matching fewer, and how much buying several tickets really helps. The wrong answers are built from typical mistakes, such as counting ordered draws (permutations) instead of combinations, or adding the bonus pool instead of multiplying.
Choose a question type or leave it mixed, select your answer to check it and read the step-by-step solution. When you want to calculate the odds of a real game, use the lottery odds calculator, which has presets for Powerball, Mega Millions, EuroMillions and UK Lotto.
How to use the Lottery Odds Quiz
- Question type
- Your answer
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Every question is generated from random game parameters, the correct answer is computed with exact combinations and the distractors come from common errors. Changing your answer keeps the question; “New question” or a different question type draws a new one.
Worked example
Question: “In a 5/69 lottery, what are the odds of matching exactly 4 of your 5 numbers?” Choose which 4 of your numbers match: C(5, 4) = 5. The fifth drawn ball must be one of the 64 numbers you did not pick: 64 ways. That gives 320 winning tickets out of C(69, 5) = 11,238,513, so the odds are 1 in 35,120.4.
Jackpot odds for common formats
| Format | Combinations | Jackpot odds |
|---|---|---|
| 5 of 35 | 324,632 | 1 in 324,632 |
| 6 of 49 | 13,983,816 | 1 in 13,983,816 |
| 6 of 59 | 45,057,474 | 1 in 45,057,474 |
| 5 of 69 + 1 of 26 | 292,201,338 | 1 in 292,201,338 |
Formula
Jackpot odds = C(N, k) · odds of m matches = C(N, k) ÷ [C(k, m) × C(N − k, k − m)] · with a separate bonus pool of B balls: × B
Worked example
Inputs — Question type: Mixed – any type; Your answer: Show me the solution.
Results — Question = A lottery draws 5 balls from 69. You pick 5 numbers. What are your odds of matching all 5 (the jackpot)?; Choices = A) 1 in 1,564,031,349
B) 1 in 5,619,257
C) 1 in 11,238,513
D) 1 in 1,348,621,560; Worked solution = Answer C: 1 in 11,238,513. Order does not matter, so count combinations: C(69, 5) = 69! / (5! × 64!) = 11,238,513. Only one of them is the winning set, so the odds are 1 in 11,238,513..
Key terms
- Combination
- A selection where order does not matter, counted by C(n, k) = n! ÷ (k!(n − k)!).
- Permutation
- An ordered selection; there are k! times more permutations than combinations.
- Prize tier
- A level of prize defined by how many main and bonus numbers you match.
Frequently asked questions
Which branch of mathematics is used for lottery odds?
Combinatorics – counting combinations with the binomial coefficient C(n, k) – together with the hypergeometric distribution.
What are the odds of the jackpot in a 6/59 lottery?
1 in 45,057,474, the number of ways to choose 6 numbers from 59.
What are the odds of matching 3 numbers in a 6/59 game?
About 1 in 96: C(6, 3) × C(53, 3) = 468,520 winning tickets out of 45,057,474.
Is matching 5 numbers more likely than matching 3?
No – in a 6/59 game matching 5 is about 1 in 141,690, far less likely than matching 3.
Can studying past draws improve my chances?
No. Draws are independent and every combination has the same chance; lotteries are games of pure chance, not skill.
Why does order not matter?
The drawn balls are compared as a set, so 1-2-3-4-5-6 and 6-5-4-3-2-1 are the same ticket. Counting ordered draws would overstate the number of outcomes by k!.