Probability
Card Draw Odds Calculator
Count all possible hands, count hands with the exact requested target-card total, and convert the ratio to probability and expected trials.
Total possible hands-
Hands with exact desired target count-
Probability of exact desired count-
Hands with zero target cards-
Probability of at least one target-
Expected exact-count hands in trials-
Expected at-least-one hands in trials-
Target share of deck-
Decision view
Exact-hand combinatorial probability tree
Exact-hand combinatorial probability treeTarget and non-target choices join into favorable hands before the exact and complement probabilities are reconciled.
Exact scenario comparisonCards drawn per hand changes while all other entered assumptions remain constant.
| Cards drawn per hand | Total possible hands | Hands with exact desired target count | Probability of exact desired count | Hands with zero target cards | Probability of at least one target | Expected exact-count hands in trials | Expected at-least-one hands in trials | Target share of deck |
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How to use Card Draw Odds Calculator
- Enter valid deck, target, draw, and exact-target counts.
- Enter the planned independent hand count.
- Compare the exact outcome with the complement probability.
Calculator guide
Understanding Card Draw Odds Calculator
Exact card odds come from counting complete hands, not multiplying independent single-card percentages.
Calculation method
How the calculation works
Use combinations to count exact-target and zero-target hands under sampling without replacement, then convert them to hand probabilities and expected trial counts. Use combinations for the target and non-target selections, divide favorable hands by all hands, and independently count the zero-target complement.
Detailed calculation process
Count exact card hands without replacement
The default deck has 52 cards, 4 targets, a 5-card draw, exactly 1 desired target, and 100 planned hands.
General formula: N_all = C(D,n)N_k = C(T,k)C(D-T,n-k)P_k = 100N_k/N_allN_0 = C(D-T,n)P_ge1 = 100(1-N_0/N_all)E_k = HP_k/100 Choose the target cards and non-target cards separately, multiply those counts, then divide by the count of every possible hand. The at-least-one probability uses the zero-target complement.
What each symbol means
Worked substitution with the default inputs
The default exact-one probability is 29.9474%, while the probability of at least one target is 34.1158%.
Purpose-built visual
Exact-hand combinatorial probability tree
The branch diagram separates target and non-target selections before reconciling favorable hands with the full sample space.
Worked situations
Practical examples
- The default deck has 52 cards, 4 targets, a 5-card draw, exactly 1 desired target, and 100 planned hands.
- The default exact-one probability is 29.9474%, while the probability of at least one target is 34.1158%.
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 deck is assumed to be well shuffled.
- Cards are sampled without replacement within each hand.
- Repeated hands are treated as independent and impossible count combinations require valid inputs.
Reference
Key terms
- Combination
- An unordered selection count.
- Favorable hand
- A hand containing exactly the requested target count.
- Complement
- All outcomes outside the named event.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why not multiply five single-card probabilities?
Draw probabilities change after every card and the hand is unordered.
Is exact one the same as at least one?
No.
Can the expected count be fractional?
Yes; it is a long-run average.
Does card order matter?
Not for the combination model.