When to use this
Use this probability model when the setup matches
Card draws, batch sampling, selecting people from a group, and any without-replacement probability question.
What this result means
Interpret the probability in three formats
The result is the probability of the requested success count range when the population size, success states, and draw count are all fixed in advance.
Formula: P(X = k) = [C(K, k) C(N-K, n-k)] / C(N, n)
Inputs
Describe the population, success states, and draws
This calculator uses a without-replacement model, so one draw changes the contents of the next draw.
Population size
The full number of items in the population before drawing.
What to enter: A positive whole number.
How to use it: Use the total group size at the start.
Example: A standard deck has population size 52.
Success states in population
How many items in the population count as successes.
What to enter: A whole number from 0 up to the population size.
How to use it: Count the target items before any draws happen.
Example: The 4 aces are the success states in a 52-card deck.
Number of draws
How many items are drawn from the population.
What to enter: A whole number from 0 up to the population size.
How to use it: Use the actual number of cards or objects drawn.
Example: A 5-card hand means 5 draws.
Target successes
The number of successes you are asking about in the sample.
What to enter: A whole number that fits the sampling setup.
How to use it: Pair it with exactly, at least, or at most.
Example: Exactly 2 aces means 2 target successes.
Event type
Choose exactly, at least, or at most.
What to enter: Deduce the event type from the wording of the question.
How to use it: At least and at most sum several valid success counts.
Example: At least 1 ace in 5 cards sums the probabilities for 1, 2, 3, and 4 aces.
Worked examples
Quick checks with common probability questions
Exactly 2 aces in 5 cards
A standard 52-card deck, 4 aces, 5 draws, and no replacement.
The probability of exactly 2 successes without replacement is 3.993%.
Load this example into the calculatorAt least 1 ace in 5 cards
This adds the probabilities for 1, 2, 3, and 4 aces in the hand.
The probability of at least 1 success without replacement is 34.1158%.
Load this example into the calculator