Calculator
Example Data Table
| Scenario | Inputs | Method | Expected Output |
|---|---|---|---|
| Award first, second, and third places from 10 people | n = 10, r = 3 | Permutation without replacement | 720 ordered outcomes |
| Choose 4 students from 12 students | n = 12, r = 4 | Combination without replacement | 495 unordered outcomes |
| Create a 4 digit code using 10 digits | n = 10, r = 4 | Permutation with replacement | 10,000 ordered outcomes |
| Draw exactly 2 successes from a mixed population | N = 20, K = 6, d = 5, k = 2 | Exact success probability | Combination ratio probability |
Formula Used
Permutation without replacement: P(n, r) = n! / (n - r)!
Permutation with replacement: n^r
Combination without replacement: C(n, r) = n! / [r! × (n - r)!]
Combination with replacement: C(n + r - 1, r)
Probability: P(Event) = Favorable outcomes / Total outcomes
Complement: P(Not event) = 1 - P(Event)
Exact success probability: [C(K, k) × C(N - K, d - k)] / C(N, d)
How to Use This Calculator
Select the calculation type first. Use permutation when order matters. Use combination when order does not matter. Select the replacement rule. Enter total items and chosen items. Add favorable outcomes to compute probability. Use manual total outcomes when your total sample space is already known. For exact success probability, enter population values and desired successes.
Statistics Guide
Why This Calculator Helps
Probability questions often mix counting and chance. A small change in wording can change the whole method. Ordered prizes need permutations. Unordered teams need combinations. Repeated draws may need replacement rules. This calculator brings those choices into one clear workflow.
Counting Before Probability
Most probability tasks start with total possible outcomes. When order matters, every position has value. A password, rank list, race finish, or seating plan usually uses permutations. When order does not matter, the same items form one group. Committees, card hands, sample groups, and lottery picks usually use combinations.
Replacement Rules Matter
Replacement changes the available pool after each choice. Without replacement, selected items leave the pool. With replacement, each pick can use the full set again. This affects both counts and probability. The calculator lets you compare these cases without rewriting formulas.
Advanced Probability Checks
The favorable outcomes field turns a count into probability. You can enter known winning cases, successful arrangements, or accepted selections. The tool divides favorable outcomes by the total count. It also reports percentage, complement probability, and odds against the event.
Hypergeometric Use
The exact success model handles draws without replacement. It is useful for quality checks, survey samples, genetics examples, card problems, and inventory testing. Enter population size, population successes, draw size, and desired successes. The calculator builds the combination ratio automatically.
Learning Value
Students can use the step details to see why a method was selected. Teachers can create practice examples quickly. Analysts can test sample space assumptions before building a report. The result table is also ready for export, so answers can be saved for worksheets or notes.
Practical Tips
Read the problem slowly. Look for words like arrange, rank, choose, select, draw, replace, and exactly. Decide whether order matters first. Then decide whether replacement is allowed. Finally compare the event count with the possible count. This simple order prevents many common mistakes.
Export And Review
Use the CSV button for spreadsheets. Use the PDF button for printable notes. Keep the input values with the answer. This makes later checking easier. Large counts may appear in scientific notation. That is normal when sample spaces grow very quickly. Review exported rows before sharing final statistical conclusions.
FAQs
What is the difference between permutations and combinations?
Permutations count ordered outcomes. Combinations count unordered outcomes. Use permutations for rankings, passwords, seating, and arrangements. Use combinations for teams, groups, samples, and selections where item order has no meaning.
When should I use replacement?
Use replacement when the same item can be selected again. Examples include digit codes, repeated dice rolls, and repeated sampling from a refreshed set. Do not use replacement when chosen items are removed.
How is probability calculated here?
The calculator divides favorable outcomes by total outcomes. Total outcomes can come from permutation counts, combination counts, or a manual value. The result also shows percentage and odds against.
What does favorable outcomes mean?
Favorable outcomes are the cases that satisfy your event. For example, winning tickets, accepted codes, target arrangements, or selected groups may be favorable. They must match the same counting method as the total outcomes.
Why is my probability greater than 1?
A probability greater than 1 means the favorable count is larger than the total count. Check whether you used the correct order rule, replacement rule, and total outcomes value.
What is exact success probability?
Exact success probability uses the hypergeometric model. It finds the chance of getting exactly k successes in d draws from a population of N items with K successes, without replacement.
Why do large answers use scientific notation?
Permutation and combination counts can grow very fast. Scientific notation keeps very large values readable. It is useful for sample spaces involving many items or long arrangements.
Can I export the results?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for printable reports, homework notes, or classroom handouts. The export uses the result table shown on the page.