Binomial Distribution Calculator: Exact and Cumulative Probability
Use this binomial distribution calculator to find exact binomial probability, cumulative probability, at least or at most probabilities, between-success ranges, mean, variance, standard deviation, and Excel-style BINOM.DIST checks. It is useful for coin tosses, pass/fail questions, survey outcomes, quality checks, and AP statistics homework.
- Exact probability Calculate P(X = k) for a fixed number of successes across independent Bernoulli trials.
- Cumulative tails Find P(X <= k), P(X < k), P(X >= k), or P(X > k) without using a printed table.
- Range probability Calculate the probability of getting between two success counts, including both endpoints.
- Distribution table Review the mean, variance, standard deviation, and a probability table for the distribution.
Calculate Binomial Probability
Enter the number of trials, the probability of success on one trial, and the success count you care about. The calculator assumes fixed independent trials with the same probability of success each time. Use decimal form for p, such as 0.25, 0.50, or 0.80.
Inputs
Results
For 12 trials with p = 0.5, the probability of exactly 7 successes is about 19.34%.
The highlighted bar shows k = 7.
| Formula | P(X = 7) = C(12, 7) * 0.5^7 * 0.5^5 |
|---|---|
| Probability | P(X = 7) = 0.1934 |
| Interpretation | About 19.34% of repeated experiments would have exactly 7 successes. |
Binomial Probability Table
The table shows exact probabilities and cumulative probabilities for the current n and p values.
| Successes k | P(X = k) | P(X <= k) | P(X >= k) |
|---|
Binomial Distribution Formula
The binomial formula counts the number of ways to arrange k successes in n trials, then multiplies by the probability of that exact arrangement.
| Exact probability | P(X = k) = C(n, k) * p^k * (1 - p)^(n - k) |
|---|---|
| Mean | mean = n * p |
| Variance | variance = n * p * (1 - p) |
| Standard deviation | standard deviation = sqrt(n * p * (1 - p)) |
Common Mistakes
- Using a binomial model when the trials are not independent.
- Using a binomial model when the probability of success changes from trial to trial.
- Mixing up at most with fewer than, or at least with more than.
- Entering the success probability as 50 instead of 0.50 when the calculator expects decimal form.
How to Calculate Binomial Probability Step by Step
Binomial homework usually becomes easier once you identify n, p, k, and the wording of the probability question.
- Check that the situation has a fixed number of trials.
- Confirm each trial has only two outcomes, usually success and failure.
- Confirm trials are independent and the success probability p stays the same.
- Use P(X = k) when the problem asks for exactly k successes.
- Use cumulative probability for phrases like at most, fewer than, at least, or more than.
- Use the between mode when the question asks for a range of success counts.
- Write the final answer as both a decimal probability and a percentage interpretation.
Binomial Probability Examples
These examples are useful for checking binomial table answers, Excel-style BINOM.DIST output, and statistics homework calculations.
| Question type | Inputs | Result | Interpretation |
|---|---|---|---|
| Exactly 7 successes | n = 12, p = 0.50, k = 7 | P(X = 7) = 0.1934 | About 19.34% of repeated experiments would have exactly 7 successes. |
| At most 7 successes | n = 12, p = 0.50, k = 7 | P(X <= 7) = 0.8062 | About 80.62% of repeated experiments would have 7 or fewer successes. |
| At least 7 successes | n = 12, p = 0.50, k = 7 | P(X >= 7) = 0.3872 | About 38.72% of repeated experiments would have 7 or more successes. |
| Between 4 and 8 successes | n = 12, p = 0.50, lower = 4, upper = 8 | P(4 <= X <= 8) = 0.8540 | About 85.40% of repeated experiments would land in this success-count range. |
When This Calculator Helps
Statistics homework
Use it when your assignment asks for exactly, at most, at least, fewer than, more than, or between probabilities for a success/failure experiment.
Real-world checks
Use it for pass/fail testing, defect counts, response rates, quiz guessing, coin flips, customer conversion counts, and other fixed-trial probability questions.
Binomial Distribution FAQs
When should I use the binomial distribution?
Use it when there is a fixed number of independent trials, each trial has success or failure, and the probability of success is the same on every trial.
What is a binomial probability?
A binomial probability is the chance of getting a certain number of successes in n independent trials when each trial has success probability p.
What is the difference between exact and cumulative probability?
Exact probability means one specific count, such as P(X = 4). Cumulative probability adds several counts, such as P(X <= 4) or P(X >= 4).
Can I use this for sampling without replacement?
Usually no. Sampling without replacement often needs a hypergeometric distribution unless the population is large enough that the binomial approximation is reasonable.
What do n, p, and k mean in a binomial problem?
n is the number of trials, p is the probability of success on one trial, and k is the number of successes you want to calculate.
What does at least mean in binomial probability?
At least k means k or more successes, so P(X >= k). At most k means k or fewer successes, so P(X <= k).
Is this like Excel BINOM.DIST?
Yes. It follows the same binomial probability ideas used in Excel-style BINOM.DIST work for exact and cumulative probabilities. Small differences can appear from rounding.
Related Statistics Resources
If the problem moves from probability to testing, interpretation, or homework write-up, these pages are good next steps.
Need Help With a Binomial Probability Problem?
Statskan can help you identify the right distribution, calculate the probability, show the formula, and write the interpretation clearly for your statistics homework.