Explore binomial results, coefficients, and outcome patterns easily. Test trials, success rates, and event ranges. See distributions, tables, exports, and explanations in one place.
Use the responsive calculator grid below. It shows three columns on large screens, two on medium screens, and one on mobile.
| Scenario | Trials (n) | Success Probability (p) | Target k | Range a-b | Exact P(X = k) | At Most P(X ≤ k) | At Least P(X ≥ k) |
|---|---|---|---|---|---|---|---|
| Email clicks in ten sends | 10 | 0.40 | 3 | 2 - 5 | 0.214991 | 0.382281 | 0.832710 |
| Quality passes in eight tests | 8 | 0.70 | 6 | 5 - 7 | 0.296475 | 0.744702 | 0.551773 |
| Correct answers in five guesses | 5 | 0.20 | 1 | 1 - 2 | 0.409600 | 0.737280 | 0.672320 |
Binomial expansion: (p + q)n = Σ C(n, x) px qn-x
Exact probability: P(X = k) = C(n, k) pk (1 - p)n-k
Cumulative probability: P(X ≤ k) = Σ from x = 0 to k of C(n, x) px (1 - p)n-x
At least probability: P(X ≥ k) = 1 - P(X ≤ k - 1)
Range probability: P(a ≤ X ≤ b) = Σ from x = a to b of C(n, x) px (1 - p)n-x
Moments: Mean = np, Variance = np(1-p), Standard deviation = √[np(1-p)]
The coefficient C(n, k) equals n! / [k!(n-k)!]. It counts how many ways k successes can appear across n trials.
Binomial probability appears in many real situations. A result can be success or failure. The number of trials is fixed. Each trial uses the same success chance. That structure fits quality checks, email clicks, quiz answers, and repeated tests. This calculator turns those inputs into fast results. It also shows the expansion logic behind each probability term. That makes the outcome easier to trust.
The calculator starts with trials, success probability, and target successes. It then evaluates the exact probability for one outcome. It also finds cumulative values. You can check at most a value, at least a value, or a full range. The page also returns the coefficient, mean, variance, and standard deviation. A distribution table lists every possible outcome. The chart helps you see where the mass sits. This is useful when comparing rare, likely, and extreme cases.
The model comes from the expansion of (p + q)^n. Here, p is the success chance. The value q equals 1 minus p. Every term in the expansion represents a possible number of successes. The coefficient counts the different arrangements. The powers of p and q weight the chance for that arrangement. This link explains why the formula works. It is not just a rule to memorize. It follows the structure of repeated independent trials.
Use this calculator for acceptance sampling, survey analysis, sports predictions, and classroom examples. It also helps in product testing. Teams can estimate defect counts in a batch. Marketers can study conversion counts from a campaign. Students can verify textbook exercises quickly. Teachers can build examples with instant tables. Analysts can export results for reports or review meetings. It saves time during review and revision work.
Keep probability values between zero and one. Enter whole numbers for trials and successes. Use realistic ranges when checking cumulative events. Read the table after every run. It often reveals patterns that one number can hide. Small shifts in p can move the peak a lot. That matters when decisions depend on thresholds. Clear inputs lead to better interpretation.
It measures exact, cumulative, and range-based binomial probabilities. It also shows the related coefficient, mean, variance, standard deviation, chart, and full distribution table.
Use it when trials are fixed, outcomes are only success or failure, each trial is independent, and the success probability stays constant across all trials.
The coefficient counts how many distinct ways k successes can occur in n trials. It is the arrangement count inside the binomial expansion and probability formula.
The value p represents success probability. The value q equals 1 minus p and represents failure probability. Both appear in every binomial term.
At least means the number of successes is k or higher. The calculator computes it with the complement rule: 1 minus the probability of getting fewer than k successes.
Convert percentages to decimals before entering them. For example, 35% should be entered as 0.35, and 80% should be entered as 0.80.
The chart reveals the distribution shape. You can quickly see the most likely outcomes, the spread, and whether the probability mass sits near the center or edges.
Avoid probabilities below zero or above one, non-whole success counts, and ranges outside the trial count. Also make sure the range end is not below the range start.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.