The mode is not the mean
A discrete distribution can have expected count between integers while its most likely row is a whole count.
Probability
Cards Outcome Table Calculator
Generate a complete exact table for every feasible target-card count, cumulative probability and count-specific score.
CARD OUTCOME DISTRIBUTION
Enumerate every feasible target-card count instead of hiding the distribution behind one percentage
This calculator constructs the full hypergeometric outcome table for one hand. Each row shows the count, combination factors, exact and cumulative probability, a count-specific net score, and its contribution to expected score. It is useful when the decision depends on the shape of all outcomes rather than one tail.
Expected target-card count-
Most likely target count-
Probability of modal count-
Expected net score-
Probability of any target-
Outcome-count entropy (bits)-
CARD OUTCOME DISTRIBUTION
Complete card-count outcome and score table
Use the table when modal, tail, cumulative, or score outcomes matter differently. Expected score is an average over repeated identical hands and does not describe volatility, utility, bankroll, or a guarantee.
| Target cards drawn | Target-card combinations | Other-card combinations | Exact probability (%) | Cumulative probability (%) | Net score | Expected-score contribution |
|---|
CURRENT CALCULATION PROCESS
Formula, exact combinations, current substitution, and reconciliation
P(X=k)=C(K,k)C(N−K,n−k)/C(N,n); E[score]=Σ[(ak−c)P(X=k)]
Every feasible target count is enumerated. Exact combinations define row probability, cumulative probability is a running sum, and count-specific score is points per target minus the fixed hand cost.
Exact combination counts are retained before display rounding.
HOW TO USE THIS MODEL
Build an outcome table with mutually exclusive rows
- Count deck size, target cards, and hand size on the same pre-draw basis.
- Define one target property so every card is either target or other.
- Assign points per target and a fixed hand cost only if that linear score matches the real rules.
- Read exact rows, cumulative probability, mode, and expected value as different summaries.
- Preserve the full table; do not report only the favorable rows or round before summing.
CARD OUTCOME DISTRIBUTION FUNDAMENTALS
Distribution-table concepts
- Mutually exclusive outcomes
- A hand cannot simultaneously contain two different target-card counts, so row probabilities add.
- Complete support
- Rows begin at the minimum feasible count and end at the maximum feasible count.
- Mode
- The row with greatest probability is the most likely count, not a guaranteed count.
- Expected count
- Probability-weighted average of target-card count over many repeated hands.
- Cumulative probability
- Running probability that X is no greater than the current row count.
- Expected score
- Sum of each row score multiplied by its probability.
MODEL AND FORMULA
Enumerate the exact hypergeometric distribution
P(X=k)=C(K,k)C(N−K,n−k)/C(N,n); E[score]=Σ[(ak−c)P(X=k)]
Every feasible target count is enumerated. Exact combinations define row probability, cumulative probability is a running sum, and count-specific score is points per target minus the fixed hand cost.
DEEPER PROBABILITY ANALYSIS
What a complete table reveals
Linear score makes expectation simple
For score ak−c, E[score]=aE[X]−c. The row sum provides an independent reconciliation of that shortcut.
Entropy describes spread, not value
Outcome-count entropy increases when probability is distributed across more plausible counts; it says nothing about whether those outcomes are desirable.
WORKED DECISION CASES
When the whole distribution matters
Suit-count scoring rule
A five-card hand receives points for each heart and pays a fixed entry score. The table shows every possible heart count and its contribution to expected score.
Designing a teaching example
An instructor changes target-card count and hand size to show how finite-population support, mode, mean, and cumulative probability move together.
TECHNICAL LANGUAGE
Outcome-table vocabulary
- Outcome row
- One exact target-card count and its associated probability.
- Modal count
- Target-card count with the highest probability.
- Expected value
- Probability-weighted average across all mutually exclusive outcomes.
- Cumulative distribution
- Probability of observing a value at or below each row.
- Entropy
- Information measure of how dispersed row probabilities are.
- Linear scoring rule
- Outcome score formed as points times count minus a constant cost.
EVIDENCE AND DATA LINEAGE
Preserve the full outcome specification
Retain deck composition, target definition, draw rule, hand size, scoring rule, fixed cost, combination counts, unrounded row probabilities, cumulative totals, and any removed or known cards. A changed scoring rule can reuse probabilities but creates a new decision table.
LIMITS AND EXCLUSIONS
Boundaries of the enumerated table
- The model handles one target/not-target partition sampled without replacement.
- The scoring rule is linear in target count; pair bonuses, sequences, ranks, suits, wild cards, and poker-hand hierarchy need explicit enumeration.
- Expected score excludes repeated-hand dependence, betting strategy, variance of monetary wealth, utility, table rules, and imperfect shuffling.
RELIABLE SOURCES
References for the finite-population model
FREQUENTLY ASKED QUESTIONS
Card outcome-table questions
Why can expected target count be non-integer?
It is a long-run probability-weighted average, not the count in one hand. Individual hands still have whole-card counts.
Why does the table sometimes start above zero?
If the hand is larger than the number of non-target cards, at least one or more targets are forced. The support reflects that constraint.
Does cumulative probability show at least k?
No. This table accumulates from the minimum through k, giving P(X≤k). An upper tail is one minus the cumulative probability below the threshold.
Can points per target be negative?
Yes for a linear penalty model. Confirm that the actual rule is truly additive and uses the same fixed cost for every outcome.
What does entropy mean here?
It summarizes dispersion among target-count outcomes in bits. It is not confidence, expected score, risk tolerance, or randomness quality.
Why keep combination counts if probabilities are shown?
They expose the exact numerator structure and make each probability auditable against the total number of possible hands.
IMPORTANT PROBABILITY NOTE
An exact table is not a promise of an outcome
The distribution is exact only for the entered finite-deck model and random without-replacement draw. Verify real rules and known cards. Expected score and probability do not guarantee profit or justify wagering.