Card Draw Probability Calculator

What this generator does

Handles draws where the pool shrinks. The lottery odds tools on this site cover the special case where the number of winning numbers and the size of your ticket are the same; a deck of 52 with 4 aces and a 5-card hand has three independent sizes and needs the general form. The result is an exact fraction computed with arbitrary-precision integers, not a floating-point approximation, and the full distribution is shown so you can see the whole shape rather than one number.

How to use this tool

1. Set the size of the deck, bag or pool.
2. Set how many of those are the ones you want.
3. Set how many you draw, and how many you need.
4. Choose at least or exactly, and read the odds.

Understanding the controls

Cards in the deck

The whole pool — a 52-card deck, a 60-card constructed deck, a bag of raffle tickets, a population you are sampling from.

How many are the ones you want

The successes in the pool: four aces, three copies of a card, the winning tickets. Everything else in the pool is a non-success.

How many you draw

Drawn together, without putting any back. That last part is the whole difference between this and a repeated single-card chance.

At least / exactly

'At least one' is almost always the question people mean, and it is not the same as 'exactly one' — on a five-card hand from a standard deck the two differ by about three percentage points.

Worked examples

5 cards from 52, 4 aces, at least 1

About 34.1% — not the 28% a with-replacement guess gives.

Exactly one ace

3243/10829, or about 29.9%.

All four aces in five cards

1 in 54,145.

The expected number of aces

5 × 4 ÷ 52 = 5/13 — the same as with replacement, because the correction is to the spread, not the mean.

Common use cases

• Working out opening-hand odds for a deck-building game
• Checking the chance of drawing a particular prize from a raffle drum
• Deciding how many copies of a card a deck needs to see one reliably
• Sampling questions — the chance a random sample contains at least one of something
• Teaching the difference between drawing with and without replacement

How this generator works

The probability of drawing exactly k of the ones you want is the number of ways to choose k from the successes, times the number of ways to fill the rest of the hand from the non-successes, over the number of possible hands. Every count is an exact integer computed with arbitrary precision, and the terms are combined on a common denominator so the cumulative 'at least' figure stays exact rather than accumulating rounding error. The distribution shown always sums to exactly one, which is asserted in the tests across a range of deck shapes.

Randomness and fairness

Nothing here is random. It is a calculator, and the same inputs always give the same exact fraction. The point of the page is that the answer is counted rather than sampled — if you want an actual draw, the card and deck tools on this site use the browser's secure generator.

For how randomness is produced across the whole site, see how Generate Random works.

Assumptions this tool makes

• The pool is well shuffled and every card is equally likely to be drawn.
• All the cards are drawn at once, with none replaced.

Limitations and good to know

• It models one draw of a fixed size. Drawing a card, looking, then drawing again with a decision in between is a different and harder question.
• Mulligans, scrying, searching and any other deck manipulation are not modelled.
• There is one category of wanted card; 'at least one ace and at least one king' needs a different calculation.
• Pool sizes are capped at 500 to keep the exact arithmetic quick.
• Nothing here is gambling advice, and knowing the odds does not change them.

Common mistakes

Multiplying the single-card chance by the number of draws

That treats the draws as independent, and they are not — every card taken changes the deck. It also breaks completely once the numbers get large, because it can exceed 100%.

Using 1 − (miss chance) to the power of the hand size

That is the with-replacement answer. It is close for a large pool and a small hand, and noticeably wrong for a small pool.

Reading 'exactly one' when you meant 'at least one'

They are different questions. At least one includes the cases where you draw two, three or four.

Practical tips

• For deck building, try a few copy counts and watch where 'at least one in the opening hand' crosses the threshold you care about.
• The expected number is a useful sanity check: draws times successes over the pool, and it is the same figure with or without replacement.
• The full distribution is often more informative than the single number — the tail is where the surprises are.

Privacy and your data

Everything is computed in your browser from four numbers. There is nothing to upload, nothing is stored, and none of your settings appear in the page address. Analytics records only that the tool was used.

Frequently asked questions

Why is this not just the single-card chance repeated?

Because the deck changes as you draw. Four aces in fifty-two is 7.7% for the first card, but if it misses, the second card is drawn from fifty-one with four aces still in it — a slightly better chance. Compounding those correctly is exactly what the hypergeometric formula does.

Is this the same as the lottery odds calculator?

It is the general case of the same distribution. A lottery is the special case where the number of winning numbers and the size of your ticket are equal; a card draw usually has three different sizes.

Can I use it for raffles or sampling?

Yes — the maths is identical. Tickets in the drum, winning tickets, tickets you bought, and how many you want to win.

Are the fractions exact?

Yes. The counts are computed with arbitrary-precision integers and reduced, so what you see is the actual fraction rather than a rounded decimal converted back.