P(X = 1)
29.947364%
Exact probability for sampling without replacementHow this calculator helps
Calculate exact or cumulative probabilities for drawing a chosen number of successes from a finite population without replacement. This page keeps its calculation method visible and groups only close keyword variants that use the same inputs and result. Use it for a quick, transparent estimate, then verify consequential decisions against the relevant current source or measurement.
How to use it
- 1
Enter your measured values in the units shown beside each field.
- 2
Review the formula and confirm that the inputs match the task you are solving.
- 3
Calculate the result, then adjust one input at a time to compare scenarios.
- 4
Use the limitations and verification guidance before ordering materials or interpreting a score.
Formula and methodology
P(X = x) = C(K,x) × C(N−K,n−x) ÷ C(N,n), where N is population size, K is the number of successes and n is the draw count.
The calculator applies the disclosed heuristic to values entered on this device. It does not load private admissions data or claim to reproduce an institution's review process.
Worked calculation example
Drawing 5 cards from a 52-card deck can be modeled with N=52, K=4 for aces and n=5.
How to interpret your result
Calculate exact or cumulative probabilities for drawing a chosen number of successes from a finite population without replacement. Read the main result together with the supporting values rather than treating one number as a guarantee. The disclosed method is: P(X = x) = C(K,x) × C(N−K,n−x) ÷ C(N,n), where N is population size, K is the number of successes and n is the draw count.
Input guide
Enter your measured values in the units shown beside each field.
Review the formula and confirm that the inputs match the task you are solving.
Calculate the result, then adjust one input at a time to compare scenarios.
Scenario comparison
| Scenario | What it demonstrates |
|---|---|
| Worked baseline | Drawing 5 cards from a 52-card deck can be modeled with N=52, K=4 for aces and n=5. |
| Input quality check | The result depends on population size, success count, sample size, target count and whether sampling is without replacement. Measure or source the values before entering them. |
| Units and conventions stress test | Keep units consistent. Different pattern, construction, academic or statistical conventions can produce different results. |
Common mistakes to avoid
- Ignoring input quality: The result depends on population size, success count, sample size, target count and whether sampling is without replacement. Measure or source the values before entering them.
- Overlooking units and conventions: Keep units consistent. Different pattern, construction, academic or statistical conventions can produce different results.
- Failing to test rounding: Displayed numbers are rounded for readability. Keep unrounded values for intermediate steps and final checks.
Check the formula, weights and rounding method published by the relevant institution before using the result officially.
Search intent and calculator coverage
Use the hypergeometric distribution when a fixed-size sample is drawn without replacement from a finite population. The page reports exact-count and at-most probabilities plus mean and standard deviation.
Related questions this calculator covers
- hypergeometric calculator
- hypergeometric probability calculator
- sampling without replacement probability
Worked scenarios
| Scenario | What it shows |
|---|---|
| Cards | count aces in a five-card hand |
| Quality sampling | estimate defect counts in a finite batch |
| Classroom problem | compare exactly x with at most x successes |
Common mistakes
- Using a binomial model when draws are without replacement
- Entering successes greater than population size
- Mixing P(X = x) with cumulative P(X ≤ x)
Check that population, success count and draw count match the experiment; compare a small case with direct combinations.
Uses the hypergeometric probability mass function and its finite-population mean and variance.What can affect the result?
Input quality
The result depends on population size, success count, sample size, target count and whether sampling is without replacement. Measure or source the values before entering them.
Units and conventions
Keep units consistent. Different pattern, construction, academic or statistical conventions can produce different results.
Rounding
Displayed numbers are rounded for readability. Keep unrounded values for intermediate steps and final checks.
Verification
Compare the result with the disclosed formula and a current primary source or qualified professional when accuracy affects a material decision.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
This hypergeometric is a browser-based estimate, not an official score, engineering design, purchase order or substitute for a tested pattern. It cannot inspect your materials or validate source measurements. Important variables include population size, success count, sample size, target count and whether sampling is without replacement.
Sources and review information
Frequently asked questions
What does the hypergeometric calculate?＋
Calculate exact or cumulative probabilities for drawing a chosen number of successes from a finite population without replacement.
How do I get a more accurate result?＋
Use careful measurements and check population size, success count, sample size, target count and whether sampling is without replacement. Keep the units consistent with the fields.
Why can another calculator give a different answer?＋
Tools can differ in assumptions, unit conversions, rounding and the exact method they use.
Does SolvePilot save the values I enter?＋
No account or personal details are needed. The calculation runs in your browser.
Can I treat this result as official?＋
No. It is a transparent estimate or practice calculation. The relevant authority, course provider, pattern maker or qualified professional remains the source of truth.
How can I check the formula?＋
Use the formula shown on this page and verify the assumptions that affect population size, success count, sample size, target count and whether sampling is without replacement.