Card Draw Probability Calculator

Work out card draw probability instantly with clear inputs, formula shown and shareable results.

About Card Draw Probability Calculator

With replacement the draws are independent so the miss chance is (1 - K/N) to the power of the draws; without replacement the hypergeometric form C(N-K, d) / C(N, d) applies. The two models diverge fast on a small deck, which is why deck-building games are analysed hypergeometrically rather than with simple independent probabilities.

Formulas

With replacement

P(no hit) = (1 - K / N) ^ draws

Without replacement

P(no hit) = C(N - K, draws) / C(N, draws)

Worked example

With the default values below, Card Draw Probability Calculator returns Chance of at least one hit of 57.8125%. Change any input above and every figure here is recalculated in your browser.

Inputs used

Cards in the deckminimum 2: 52
Cards that count as a hitminimum 1: 13
Cards drawnminimum 1: 3
Return each card before the next draw: Yes

Results

Chance of at least one hit: 57.8125%
Chance every draw is a hit: 1.56250%
Chance of no hits: 42.1875%
Hits expected: 0.750

Step by step

1Values used
2With replacement
3Without replacement
4Chance of at least one hit= 57.8125
5Chance every draw is a hit= 1.56250
6Chance of no hits= 42.1875
7Hits expected= 0.750

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is the Card Draw Probability figure worked out?

P(no hit) = (1 - K / N) ^ draws. With replacement the draws are independent so the miss chance is (1 - K/N) to the power of the draws; without replacement the hypergeometric form C(N-K, d) / C(N, d) applies.

Why does this matter?

The two models diverge fast on a small deck, which is why deck-building games are analysed hypergeometrically rather than with simple independent probabilities.

What do I need to enter?

Enter cards in the deck, cards that count as a hit, cards drawn, return each card before the next draw. The defaults shown are a realistic worked example — swap in your own figures to get a result you can use.