Calculator Inputs
Model custom decks, removed cards, jokers, exact hits, lower bounds, upper bounds, and full hypergeometric draw distribution.
Example Data Table
| Scenario | Deck Size | Hand Size | Favorable Cards | Target | Condition |
|---|---|---|---|---|---|
| Poker flush study | 52 | 5 | 13 | 5 | Exactly |
| Face card hit | 52 | 5 | 12 | 2 | At least |
| Red card sample | 52 | 7 | 26 | 3 | At most |
| Rare combo deck | 40 | 6 | 4 | 0 | None |
Formula Used
The total number of possible hands is computed with the combination formula:
C(n, r) = n! / (r! × (n − r)!)
Here, n is the effective deck size and r is the number of cards drawn.
For favorable events, the calculator uses the hypergeometric model:
P(X = x) = [C(K, x) × C(N − K, n − x)] / C(N, n)
N is the effective deck size, K is the number of favorable cards in the deck, n is the draw size, and x is the number of favorable cards drawn.
For “at least” and “at most” cases, the calculator sums relevant exact probabilities over the allowed range.
How to Use This Calculator
- Enter the base deck size for your card system.
- Add jokers if your game includes extra wild cards.
- Enter removed cards if some cards are excluded already.
- Set the number of cards drawn into the hand.
- Enter how many cards count as favorable outcomes.
- Choose the target favorable cards and the probability condition.
- Press calculate to see total combinations, event count, probability, and full distribution.
- Use the CSV or PDF buttons to export your results.
Frequently Asked Questions
1. What does this calculator actually measure?
It measures how many distinct card hands are possible and how likely a chosen event is. Events include exact matches, minimum hits, maximum hits, or zero favorable cards in a draw.
2. What are favorable cards?
Favorable cards are the cards you define as success cards. They might be hearts, face cards, aces, combo pieces, wild cards, or any subset relevant to your math problem.
3. Why use combinations instead of permutations?
Most card hands ignore order. A hand containing the same cards is considered identical regardless of draw sequence, so combinations give the correct count for unordered selections.
4. What is the effective deck size?
Effective deck size equals base deck size plus jokers minus removed cards. This lets you model shortened decks, custom card sets, or games where certain cards are excluded beforehand.
5. What does the distribution graph show?
The graph shows the probability for every possible number of favorable cards drawn. It helps you see where the most likely outcomes sit and how quickly probabilities drop away.
6. Can this calculator help with poker or trading card games?
Yes. It works well for poker studies, collectible card games, classroom probability exercises, and any deck-based system where draws happen without replacement from a finite pool.
7. What does “at least” mean here?
“At least” means the event includes the chosen target and every larger feasible favorable count. For example, at least 2 includes outcomes with 2, 3, 4, and so on.
8. Why are expected value and standard deviation included?
They summarize the center and spread of the draw distribution. Expected value gives the average favorable count, while standard deviation shows how widely outcomes vary around that average.