For understanding probability exercises
How to calculate binomial probability
Start with a fixed trial count, one success probability and the event in your question. Follow a worked example from n, p and k to a checked probability, then calculate your own values below.
Work through the exampleFree guide. Calculate your own values on this page, without uploading them.
First check that the binomial model fits
You need a fixed number of trials, two outcomes on each trial, independence and the same success probability each time. Choose what “success” means before assigning numbers. The other outcome is “failure,” regardless of whether either outcome is desirable.
If a question samples without replacement from a small finite group, the probability can change from draw to draw. Do not automatically use a binomial model in that situation. This calculator checks your numbers, but it cannot verify the assumptions in a written problem.
A worked example from question to answer
This is a constructed mathematical exercise. Suppose there are five independent trials, each with a 20% chance of success. Find the probability of exactly two successes.
1. Translate the question into n, p and k
- The number of trials is
n = 5. - The per-trial probability is
p = 0.2. Convert 20% by dividing by 100. - The requested success count is
k = 2.
2. Write down the event
“Exactly two” means X = 2. It does not include zero, one or three successes. Choose the PMF for this event. If the question says “at most” or “at least,” first identify all included counts.
3. Substitute into the PMF formula
P(X = 2) = C(5,2) × 0.2² × 0.8³
Compute the combination count C(5,2) = 5! / (2! × 3!) = 10. There are ten ways to place two successes among five trials. Each arrangement has probability 0.2² × 0.8³ = 0.02048.
4. Multiply and convert to a percentage
P(X = 2) = 10 × 0.02048 = 0.2048. Multiplying by 100 gives 20.48%. The probability is a number between 0 and 1; the percentage is the same result on a 0 to 100 scale.
To check it below, enter 5, 0.2 and 2, leave Exactly k selected, confirm the assumptions and calculate. The result should be 0.2048. The displayed example stays separate from your own calculation.
If the question says at least two
Keep n = 5 and p = 0.2, but change the event. At least two includes counts 2, 3, 4 and 5. You can add four PMF terms, or use the complement of zero or one success.
P(X≥2) = 1 − P(X=0) − P(X=1)
= 1 − 0.32768 − 0.4096
= 0.26272 = 26.272%
Exactly two and at least two answer different questions. The event-boundary guide explains which counts each phrase includes. The CDF calculator is useful for checking cumulative events.
Check the answer before using it
- Confirm that n counts trials and k counts successes, not the other way around.
- Enter the success probability for one trial, not the final probability you are trying to find.
- Read the event next to the result. A correct calculation for the wrong inequality is still the wrong answer to your question.
- Keep unrounded values while working. Round at the end, according to the precision your exercise requests.
If a probability is extremely small, scientific notation keeps it distinct from zero. You can inspect the PMF formula and combination count, plus the full distribution table after calculating.
Formula reference NIST’s binomial distribution handbook. The numerical examples above were computed independently with integer and decimal arithmetic.
Check your own probability
Enter the values from your question. Pick its wording, confirm the assumptions and calculate. You can then open the working, statistics, table or graph. Editing a value clears the previous result.
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Find the explanation you need
Exactly k successes, with combinations and formula steps.
Cumulative probability, tail boundaries and inverse cutoffs.
The one-count difference between ≥ and >, with examples.
All tools use 0 ≤ n ≤ 1000 and decimal p from 0 to 1. They do not choose a model or interpret a written problem automatically. Read privacy, assumptions and use limits.