RandoziDraw Without Replacement Odds
Put red and blue items in an urn, draw some without putting them back, and see the exact chance of every number of reds: as a reduced fraction, as a percentage, and “at least” or “at most”.
Chance of exactly 2 red
3/10
30 %
| Reds | Exactly | At least | At most |
|---|---|---|---|
| 0 | 1/10 (10 %) | 1/1 | 1/10 |
| 1 | 3/5 (60 %) | 9/10 | 7/10 |
| 2 | 3/10 (30 %) | 3/10 | 1/1 |
10 equally likely ways to draw 2 from 5 (up to 1000 items).
How it works
- Enter the number of red items and blue items in the urn.
- Enter how many you draw, without putting any back.
- Read the table: the chance of exactly, at least and at most each number of reds.
- Copy the table to share the odds.
What people use it for
Raffles and draws
You hold 3 of 50 tickets and 2 prizes are drawn: what is the chance you win at least one?
Card and board games
What are the odds of two aces in a five-card hand taken from the 4 aces and 48 other cards?
Classroom and statistics
Check a hypergeometric exercise with exact integers, not rounded decimals.
Questions
- What is the formula?
- The chance of exactly k reds in n draws is C(R, k) × C(B, n − k) ÷ C(R + B, n), where C counts combinations, R is the number of reds and B the number of blues. With 3 reds and 2 blues and 2 drawn, both are red in 3 of 10 ways: 3/10.
- Why exact fractions?
- The counts use whole numbers of any size, so 3/10 is exactly 3/10 and not 0.30000000000000004. The percentage is rounded to 4 decimals for display only.
- What if I put the items back?
- Then the draws are independent and the odds are different (a binomial law). This page is for draws without replacement, where each draw changes what is left.
- Are there limits?
- Up to 1,000 items in the urn. You cannot draw more than the urn holds, and a count that cannot happen (for example 3 reds from an urn with 2) shows no row.