CTCD

Probability

Card Target Count Distribution Calculator

Build the exact probability distribution for zero through the displayed maximum target cards in a hand.

Deck sizeTarget cards in deckCards drawn per handMaximum target count displayedPlanned hands

Total possible hands-

Probability of zero targets-

Probability every drawn card is target-

Expected target count per hand-

Expected total target cards across hands-

Displayed distribution rows-

Decision view

Hypergeometric target-count summary

Hypergeometric target-count summaryZero-target and all-target probabilities frame the entered hand while the expected target count provides the central reference.

Exact scenario comparisonTarget cards in deck changes while all other entered assumptions remain constant.

Target cards in deck	Total possible hands	Probability of zero targets	Probability every drawn card is target	Expected target count per hand	Expected total target cards across hands	Displayed distribution rows

Period-by-period detail

Exact target-card count probability distribution

Every feasible displayed target count is calculated from the complete hypergeometric hand count.

How to use Card Target Count Distribution Calculator

Enter deck, target, and draw counts.
Choose the maximum displayed target count.
Inspect the probability histogram and exact table.

Calculator guide

Understanding Card Target Count Distribution Calculator

A complete target-card distribution assigns an exact hypergeometric probability to every feasible target count.

Count the sample space Every five-card combination appears once.

Calculate zero targets All five cards come from the 39 non-target cards.

Calculate one target One target is the modal outcome under the default composition.

Calculate the remaining bars Every count uses the same exact combination identity.

Calculation method

How the calculation works

Generate the hypergeometric probability for every displayed target-card count using exact combinations without replacement. For each target count, multiply the target and non-target combination counts and divide by all possible hands.

Detailed calculation process

Build the complete target-count distribution

The default draws 5 cards from a 52-card deck containing 13 targets and displays counts 0 through 5 across 200 planned hands.

General formula: N_all = C(D,n)N_j = C(T,j)C(D-T,n-j)P_j = 100N_j/N_allSumP = sum(P_j)E[X] = nT/DE_j = HP_j/100 Each bar is a mutually exclusive target count. The hypergeometric terms count the exact target and non-target selections, and the complete feasible distribution should sum to 100%.

What each symbol means

D, T Deck size and target cards (cards).

n Cards drawn (cards).

j Target cards in one displayed outcome (cards).

N_j Hands with exactly j targets (hands).

P_j Probability of exactly j targets (%).

H Planned hands (hands).

Worked substitution with the default inputs

1. Count the sample space N_all = C(52,5) = 2,598,960 hands Every five-card combination appears once.

2. Calculate zero targets N_0 = C(13,0)C(39,5) = 575,757P_0 = 22.1534% All five cards come from the 39 non-target cards.

3. Calculate one target N_1 = C(13,1)C(39,4) = 1,069,263P_1 = 41.1420% One target is the modal outcome under the default composition.

4. Calculate the remaining bars P_2 = 27.4280%P_3 = 8.1543%P_4 = 1.0729%P_5 = 0.0495% Every count uses the same exact combination identity.

5. Reconcile mean and total E[X] = 5(13/52) = 1.25 targets/handSumP = 100.0000% The expected value and probability sum independently check the distribution.

The default distribution peaks at one target card with 41.1420%, and the expected target count is 1.25 per hand.

Purpose-built visual

Hypergeometric probability histogram

Discrete bars expose the full shape, mode, tails, and expected target count.

Live The visual is regenerated from the current inputs.

Units Counts, money, force, concentration, mass, and percentages retain their stated units.

Check The plotted values reconcile to the displayed calculation.

Worked situations

Practical examples

The default draws 5 cards from a 52-card deck containing 13 targets and displays counts 0 through 5 across 200 planned hands.
The default distribution peaks at one target card with 41.1420%, and the expected target count is 1.25 per hand.

Better inputs

Useful tips

Change one input at a time and confirm that both the results and visual update.
Keep every input in the unit printed beside it.
Retain intermediate precision and round only the reported result.

Before relying on the result

Limitations and common mistakes

The model assumes sampling without replacement.
Displayed rows must include every feasible count to reconcile to 100%.
Expected counts do not predict a particular hand.

Reference

Key terms

Hypergeometric: A without-replacement count distribution.
Probability mass: Probability assigned to one discrete outcome.
Expected value: The long-run average target count.

Important note

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

Frequently asked questions

Why are the bars discrete?

A hand can contain only whole target-card counts.

Should all bars sum to 100%?

Yes, when every feasible count is displayed.

Why is one target most likely?

It reflects the entered deck share and draw size.

Is the mean always a possible outcome?

No; an expected value can be fractional.