Lottery Odds Calculator (pick k of n, + bonus ball)

Understanding the lottery odds calculator — concept and principle

A lottery odds calculator works out the chance of winning a "pick k of n" lottery, optionally with a separately drawn bonus ball, and the chance of matching exactly three numbers. It covers 6/45 (Korea's Lotto), 6/49 and Powerball-style games.

How it is calculated

• Combinations C(n, k) = n! ÷ (k! × (n − k)!), the number of different tickets possible.
• Odds of jackpot = 1 in C(n, k) × B, where B is the size of a separate bonus pool (B = 1 if there is none).
• Probability = 1 ÷ (C(n, k) × B).
• Odds of matching 3 = C(n, k) ÷ [C(k, 3) × C(n − k, k − 3)], the chance that exactly 3 of your k main numbers are among the k drawn.

The bonus pool models a second, independent drum, as in Powerball (5 of 69 plus 1 of 26). If the bonus comes from the same pool, as in Korea's Lotto 6/45, choose that option: the jackpot stays 1 in C(n, k) and the calculator adds the odds of k − 1 numbers plus the bonus, C(n, k) ÷ k (1 in 1,357,510 for 6/45).

The principle

In a fair draw every set of k numbers is equally likely, and order does not matter, so the number of possible outcomes is the binomial coefficient C(n, k). One ticket covers exactly one of them. For partial matches, count favourable outcomes: choose which 3 of your k numbers are drawn, C(k, 3), and which k − 3 of the drawn numbers come from the n − k numbers you did not pick, C(n − k, k − 3). This is the hypergeometric distribution. A separate bonus drum multiplies the outcomes by its size.

Worked example

1. Inputs: n = 45, k = 6, no bonus pool.
2. C(45, 6) = 45 × 44 × 43 × 42 × 41 × 40 ÷ 720 = 8,145,060, so the jackpot odds are 1 in 8,145,060.
3. Probability = 1 ÷ 8,145,060 ≈ 1.2277 × 10⁻⁷.
4. Exactly 3 matches: C(6, 3) = 20 and C(39, 3) = 9,139, so 20 × 9,139 = 182,780 favourable sets.
5. Odds = 8,145,060 ÷ 182,780 ≈ 1 in 44.56.

How to read the result

Odds of 1 in 8 million mean that buying one ticket a week for 1,000 years would still give only about a 0.6% chance of a jackpot. For a Powerball-style game (69, 5, bonus 26) the calculator gives 1 in 292,201,338. Buying more tickets raises the chance in proportion, but the expected value of a lottery ticket is negative, because only part of ticket sales is paid back as prizes.

Common mistakes

• Thinking numbers that have not appeared for a while are "due"; every draw is independent.
• Entering a bonus pool size for a game whose bonus ball is drawn from the main pool; choose "the same pool" instead.

Frequently asked questions

Does choosing popular numbers change my odds?

No, but if you win you are more likely to share the prize with others who picked the same numbers.

What are the odds of matching 3 numbers in Lotto 6/45?

About 1 in 44.6 for exactly 3 matches, which is the fifth prize in Korea's Lotto.

Do two tickets with different numbers double my chance?

Yes, two distinct tickets give exactly twice the probability, still about 1 in 4 million.

Sources: Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley), hypergeometric distribution · Ross, S. M., A First Course in Probability (Pearson), combinatorial analysis