Card Probability Calculator
Calculate hypergeometric probabilities for drawing a chosen number of target cards from a finite deck without replacement.
Card Probability Calculator inputs
Use the fields below for this exact calculation. Every visible assumption can be changed before the result is interpreted.
Card Probability Calculator results
How the card probability calculator works
P(X=k)=C(K,k)C(N−K,n−k)/C(N,n); cumulative modes sum valid exact probabilities over the requested hit range.
Why card draws use a hypergeometric model
A card probability calculator for draws without replacement must update the available deck after every draw. The hypergeometric distribution performs that counting directly. It chooses the target cards and non-target cards that make a hand, then divides by all possible hands of the same size.
Defining the deck and targets
N is the total number of cards, K is the number that satisfy the target condition, and n is the number drawn. A target might be an ace, a heart, any face card, or a custom marked category. Count overlapping descriptions carefully: for example, the ace of hearts belongs to both the ace and heart groups and should not be double-counted when the target is their union.
Exactly, at least, and at most
Exactly k uses one hypergeometric term. At least k sums every feasible hit count from k upward, while at most k sums from the minimum feasible count through k. The no-target mode is the exact-zero case. This card probability calculator determines the feasible range first, so impossible hit counts contribute zero rather than causing invalid combinations.
Hands versus ordered sequences
A five-card hand containing one ace is the same hand regardless of which draw position produced the ace. Combination counts therefore ignore order. If the question depends on a particular sequence—such as the first card being an ace and the second being a king—a conditional or permutation-based calculation is required instead of this hand-count model.
Expected targets and odds against
The expected number of targets is nK/N. It is a long-run average, not the most likely exact hand and not a guarantee for one draw. Odds against are displayed as (1−p):p when the event probability is positive. Probability percentage, decimal probability, and distribution bars provide three consistent views of the same model.
Assumptions and fair use
The deck composition must be known, every eligible hand must be equally likely, and cards must not be replaced between draws. Shuffling quality, hidden removals, wild-card rules, and strategic information can change a real game. Use this card probability calculator to analyze a stated random-draw model, not to imply certainty about a particular shuffled deck.
Card Probability Calculator: reliable-use checklist
Enter the exact model, not a nearby problem
The card probability calculator is deliberately limited to the rules described on this page. Confirm that the entered relationship, timing convention, domain, units, and event definition match the original problem. A numerically precise answer can still be inappropriate when the wrong model is selected.
Keep signs, order, and precision visible
Negative signs, function order, coordinate order, draw conditions, and beginning-versus-end timing can change the result materially. Preserve full precision during an independent check and round only the reported answer. The supporting cards are included so the main result is not copied without its assumptions.
Test a known example before relying on a custom case
Load the worked example and confirm that the displayed method behaves as expected. Then change one input at a time. This exposes sign errors, invalid ranges, and misunderstood settings more effectively than replacing every default simultaneously.
Save enough information to reproduce the answer
Keep the entered values, selected options, date, result cards, and any limitation shown in the detailed report. The card probability calculator runs locally and does not create an account or store a calculation history, so reproducibility depends on the record retained by the visitor.
Card Probability Calculator questions
Does this calculator assume replacement?
No. It uses the hypergeometric distribution for draws without replacement.
What should I enter for four aces in a standard deck?
Enter 52 total cards and 4 target cards, then choose the hand size and desired number of hits.
Why is expected value not a whole number?
It is the average target count over many repeated hands, so it need not be a possible count in one hand.
Can it calculate an ordered first-card event?
Not directly. Ordered events need conditional sequence probabilities rather than combination-based hand counts.