CDP

Probability

Card Draw Probability Calculator

Calculate exact desired-target probability, at-least-one probability, expected successes, and a payoff expectation for unordered hands drawn without replacement.

Cards in deckTarget cards in deckCards drawn without replacementExact target-card count desiredPlanned independent handsStake per handNet win per successful handEntered probability comparison (%)

Possible unordered hands-

Hands with exact desired target count-

Exact desired-count probability-

Hands containing no target cards-

Probability of at least one target-

Expected exact successes-

Entered payoff expected net result-

Exact probability minus entered comparison-

Target cards as deck share-

Decision view

Card-draw probability calculation

Card-draw probability calculationDeck composition and draw size become total hands, successful hands, probability, and expected outcomes.

Exact scenario comparisonCards drawn without replacement changes while all other entered assumptions remain constant.

Cards drawn without replacement	Possible unordered hands	Hands with exact desired target count	Exact desired-count probability	Hands containing no target cards	Probability of at least one target	Expected exact successes	Entered payoff expected net result	Exact probability minus entered comparison	Target cards as deck share

How to use Card Draw Probability Calculator

Enter deck, target-card, draw, and desired counts.
Enter planned independent hands and payoff assumptions.
Compare exact and at-least-one events.
Use the probability tree and outcome composition together.

Calculator guide

Understanding Card Draw Probability Calculator

Drawing cards without replacement follows a hypergeometric model. Counting hands exactly keeps the desired target count separate from the broader chance of at least one target.

Exact event Exactly x targets.

Broader event At least one target.

No replacement Deck composition changes during a hand.

Payoff Uses exact-event probability.

Calculation method

How the calculation works

Count unordered hands exactly, separating exact desired target-card count from the broader at-least-one probability and extending only the exact case to entered payoff arithmetic. Count all combinations of draw-size cards, multiply target and non-target combinations for the exact event, and use the no-target complement for at least one.

Detailed calculation process

Count exact card-draw outcomes without replacement

The default deck contains 52 cards with four targets; five cards are drawn and exactly one target is desired across 100 independent hands.

General formula: N = C(D,n)F_x = C(T,x)C(D-T,n-x)p_x = F_x/Np_ge1 = 1-C(D-T,n)/NH = mp_xEV = H W-(m-H)c Combination counts ignore order within a hand. The exact event chooses x targets and n-x non-targets, while at least one is easiest to calculate as one minus the no-target probability.

What each symbol means

D, T Deck size and number of target cards.

n, x Cards drawn and exact desired target count.

C(a,b) Number of unordered b-card selections from a cards.

N Total possible unordered hands.

F_x, p_x Exact favorable hands and exact probability.

p_ge1 Probability of at least one target.

m, H Planned hands and expected exact successes.

W, c, EV Net win, stake, and modeled expected net result (currency).

Worked substitution with the default inputs

1. Count all five-card hands N = C(52,5) = 2,598,960 Order inside the five-card hand is irrelevant.

2. Count exact one-target hands F_1 = C(4,1)C(48,4)F_1 = 4x194,580 = 778,320 One target and four non-target cards fill the hand.

3. Calculate exact probability p_1 = 778,320/2,598,960 = 0.299474100p_1 = 29.9474% This excludes hands with two or more targets.

4. Calculate at least one p_ge1 = 1-C(48,5)/C(52,5)p_ge1 = 34.1158% The complement removes only hands with zero targets.

5. Project and reconcile payoff H = 100x0.299474 = 29.9474EV = 29.9474x$30-(100-29.9474)x$5 = $548.16 The payoff arithmetic uses the exact-one event and entered net win and stake.

The default exact-one probability is 29.9474%, the at-least-one probability is 34.1158%, and the entered payoff model yields $548.16 over 100 hands in expectation.

Purpose-built visual

Follow a hypergeometric outcome tree

A target/non-target branch diagram terminates in exact, other-success, and no-target outcome shares, all changing with deck composition.

Root Complete hand space.

Exact branch Desired target count.

Other branch Additional target counts.

No-target branch Complement event.

Worked situations

Practical examples

There are 2,598,960 unordered five-card hands.
Exactly one of four targets appears in 778,320 hands.
At least one is broader than exactly one.

Better inputs

Useful tips

Validate that desired targets do not exceed targets or draw count.
Keep net win and stake definitions consistent.
Use expected value only for repeated comparable trials.

Before relying on the result

Limitations and common mistakes

The deck is uniformly shuffled.
Hands are independent and cards are drawn without replacement.
Wild cards, strategy, conditional draws, and bankroll risk are excluded.

Reference

Key terms

Combination: Unordered selection count.
Hypergeometric: Sampling successes without replacement.
Complement: One minus the opposite event.
Expected value: Probability-weighted long-run average.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why not use 4/52 times 5?

That shortcut does not correctly handle overlap and changing deck composition.

Is exactly one the same as at least one?

No. At least one also includes two or more targets.

Why are hands unordered?

The event depends on composition, not draw order.

Is positive expected value guaranteed profit?

No. Actual repeated outcomes vary and the model omits risk constraints.