Probability
Binomial Distribution Calculator
Compute the binomial point probability, the chance of k or fewer successes, the chance of k or more, and the mean and variance.
Worksheet
Press Calculate to see the result.
Compute the binomial point probability, the chance of k or fewer successes, the chance of k or more, and the mean and variance.
What this calculator answers
A binomial count is the number of successes in a fixed run of independent trials that share one success probability. Coin flips, defect checks, and yes-or-no items on a fixed questionnaire fit that description when the trials do not change each other. This page evaluates the probability of one exact count, the probability of that count or fewer, and the probability of that count or more. It also reports the mean and variance that belong to those trials.
The point mass uses the binomial probability routine. The lower cumulative probability sums every mass from 0 through k. The upper cumulative probability is one minus the sum through k − 1, so the exact count k belongs to both cumulatives. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Binomial probability mass and cumulatives uses this relationship.
P(X = k) = C(n, k) p^k (1 − p)^(n − k), mean = np, variance = np(1 − p)
- n. The fixed number of trials.
- p. The success probability on each trial.
- k. The success count being evaluated.
- q. The failure probability, 1 − p.
Each sequence with k successes and n − k failures has probability p^k (1 − p)^(n − k), and C(n, k) counts those sequences. Adding the masses from 0 through k builds the cdf. The mean np is the long-run successes per run of n trials, and the variance multiplies that mean by the failure probability. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
There are 10 trials, the success probability is 0.5, and the count is 6. The point probability P(X = 6) is 0.2051. The lower cumulative P(X ≤ 6) adds every mass from 0 through 6 and equals 0.8281.
The upper cumulative P(X ≥ 6) equals 0.377. Both cumulatives include the mass at 6, so they are not complements and they do not add to 1. The mean is n × p = 10 × 0.5 = 5.
The variance is np(1 − p) = 5 × 0.5 = 2.5. p was checked against the closed interval from 0 to 1, and both n and k were required to be whole numbers.
| P(X = k) | 0.2051 |
|---|---|
| P(X ≤ k) | 0.8281 |
| P(X ≥ k) | 0.377 |
| Mean np | 5 |
| Variance np(1 − p) | 2.5 |
A second situation
The same 10 trials with a rarer success probability
There are 10 trials, the success probability is 0.2, and the count is 6. The point probability P(X = 6) is 0.0055. The lower cumulative P(X ≤ 6) adds every mass from 0 through 6 and equals 0.9991.
The upper cumulative P(X ≥ 6) equals 0.0064. Both cumulatives include the mass at 6, so they are not complements and they do not add to 1. The mean is n × p = 10 × 0.2 = 2.
The variance is np(1 − p) = 2 × 0.8 = 1.6. p was checked against the closed interval from 0 to 1, and both n and k were required to be whole numbers.
| P(X = k) | 0.0055 |
|---|---|
| P(X ≤ k) | 0.9991 |
| P(X ≥ k) | 0.0064 |
| Mean np | 2 |
| Variance np(1 − p) | 1.6 |
Dropping p from 0.5 to 0.2 keeps n and k fixed but makes six successes a rarer event, so the point probability falls and the mean falls from 5 to 2. The same formula is doing the work. Only the success probability changed. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The observed success count moved from 6 to 8. The headline result went from 0.2051 to 0.0439.
The nudged inputs
There are 10 trials, the success probability is 0.5, and the count is 8. The point probability P(X = 8) is 0.0439. The lower cumulative P(X ≤ 8) adds every mass from 0 through 8 and equals 0.9893.
The upper cumulative P(X ≥ 8) equals 0.0547. Both cumulatives include the mass at 8, so they are not complements and they do not add to 1. The mean is n × p = 10 × 0.5 = 5.
The variance is np(1 − p) = 5 × 0.5 = 2.5. p was checked against the closed interval from 0 to 1, and both n and k were required to be whole numbers.
| P(X = k) | 0.0439 |
|---|---|
| P(X ≤ k) | 0.9893 |
| P(X ≥ k) | 0.0547 |
| Mean np | 5 |
| Variance np(1 − p) | 2.5 |
How to read the result
Read P(X = k) as the probability of that exact count, not of a range. Read the two cumulatives with care: both include k, so they are not two pieces of a partition and they do not add to 1. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The number of trials is fixed before you observe the count.
- Each trial is a success or a failure.
- The trials are independent.
- Every trial uses the same p.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Treating P(X ≤ k) and P(X ≥ k) as complements. They overlap at k.
- Entering a percent such as 50 in the p box. The box wants 0.50.
- Using a binomial model when the draw is without replacement from a small population. That is hypergeometric.
- Letting n float. Trials and counts have to be whole numbers.
Where a different calculator is the honest choice
Use the normal approximation page when you want a continuity-corrected bell curve beside the exact sum. Use the Poisson calculator when n is large, p is small, and the mean np is the natural input. Use the negative binomial calculator when the number of successes is fixed and the number of failures is random. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Normal Approximation to Binomial Calculator, Poisson Distribution Calculator, Negative Binomial Distribution Calculator.
Check yourself
For 10 independent trials with p = 0.5, what is P(X = 6), and what are the mean and variance?
P(X = 6) is 0.2051, P(X ≤ 6) is 0.8281, P(X ≥ 6) is 0.377, the mean is 5, and the variance is 2.5.
Using the result in a sentence
Fixed sets of coin flips, pass-fail items, and quality checks with a stable defect rate are the home of this calculation. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Independence fails when one trial changes the next, as in sampling without replacement from a short list. A changing p across trials also takes the count outside this formula. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Binomial Distribution Calculator commit to?
This page uses Binomial probability mass and cumulatives: P(X = k) = C(n, k) p^k (1 − p)^(n − k), mean = np, variance = np(1 − p). Each sequence with k successes and n − k failures has probability p^k (1 − p)^(n − k), and C(n, k) counts those sequences. Adding the masses from 0 through k builds the cdf. The mean np is the long-run successes per run of n trials, and the variance multiplies that mean by the failure probability. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Trials n, Success probability p, Successes k, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the p(x = k) is 0.2051. Read P(X = k) as the probability of that exact count, not of a range. Read the two cumulatives with care: both include k, so they are not two pieces of a partition and they do not add to 1. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Treating P(X ≤ k) and P(X ≥ k) as complements. They overlap at k. Independence fails when one trial changes the next, as in sampling without replacement from a short list. A changing p across trials also takes the count outside this formula. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the normal approximation page when you want a continuity-corrected bell curve beside the exact sum. Use the Poisson calculator when n is large, p is small, and the mean np is the natural input. Use the negative binomial calculator when the number of successes is fixed and the number of failures is random. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
P(X = 6) is 0.2051, P(X ≤ 6) is 0.8281, P(X ≥ 6) is 0.377, the mean is 5, and the variance is 2.5. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.