Quiz: Lottery calculator

How the quiz works

A question appears as soon as the page loads. Type your answer in the box and press the button. The result says whether you were right, gives the correct answer, and opens the step-by-step working. Press the button again to get a new question, then type over your old answer. Changing the difficulty starts a new question at that level straight away.

The formulas you need

Tickets in a k-from-n draw: C(n, k) = n! ÷ (k! (n − k)!) Jackpot odds: 1 in C(n, k) With a bonus pool of B: 1 in C(n, k) × B Exactly m matches: C(k, m) × C(n − k, k − m) ÷ C(n, k) Expected value: jackpot ÷ C(n, k) − ticket price At least one win in t tries: 1 − (1 − 1/C(n, k))ᵗ

A worked example

An easy question might read: “A lottery draws 6 numbers from 1 to 49. You pick 6. What are the odds of matching all 6?” The order of the balls does not matter, so count combinations:

C(49, 6) = (49 × 48 × 47 × 46 × 45 × 44) ÷ 720 = 13,983,816

The answer is 1 in 13,983,816.

A medium question might ask for the odds of matching exactly 3 of the 6. You need 3 of the 6 winning numbers, in C(6, 3) = 20 ways, and 3 of the 43 losing numbers, in C(43, 3) = 12,341 ways:

20 × 12,341 = 246,820 winning tickets 13,983,816 ÷ 246,820 = 56.66 → 1 in 56.66

A hard question might give a $5,000,000 jackpot and a $2 ticket in the same game and ask for the expected value. The expected winnings are 5,000,000 ÷ 13,983,816 = $0.3576, so the expected value is $0.3576 − $2 = −$1.64 per ticket.

Another hard type asks about repeated play: with 1 ticket a week for 520 weeks (ten years), the chance of at least one jackpot is 1 − (1 − 1/13,983,816)⁵²⁰ ≈ 0.0000372. Working out the chance of never winning and subtracting it from 1 is easier than adding up the chances of winning in week 1, week 2, and so on.

Tips for getting them right

• Cancel before you multiply. C(49, 6) needs only the top six factors of 49! over 6!.
• Multiply by the bonus pool, never add it. Each main combination pairs with every bonus number.
• For exactly m matches, remember the second factor: your other numbers must all be losers.
• Subtract the ticket price once, at the end, when finding the expected value.

To check a question with different numbers, the lottery calculator gives the odds of every tier at once, and the combination calculator evaluates any C(n, k).