Binomial probability calculator: exact, at least, at most

Q: What is the difference between at least and at most on a binomial calculator?

At most k means k or fewer successes, so you add P(0) through P(k). At least k means k or more, so you add P(k) through P(n), or faster, take 1 - P(X ≤ k-1). Watch the wording, because more than 4 is P(X ≥ 5), not P(X ≥ 4).

Q: Do I enter p as a decimal or a percent?

Enter p as a decimal between 0 and 1. A 30 percent success rate is 0.3, and a 5 percent rate is 0.05. Entering 30 instead of 0.3 gives output that makes no sense.

By Jude Wallis · Published July 28, 2026

A binomial probability calculator finds the chance of a set number of successes in n independent trials, each with success probability p. Enter n, p, and x to get the exact P(X = x), or an at-least or at-most range using cumulative probabilities.

Trials (n)Success probability (p)Successes (x)

binomial probability

0.2668

Distribution summary: mean = np = 3, SD = sqrt(np(1 - p)) = 1.4491.

Steps

1.P(X = 3) = C(10, 3) x 0.3^3 x 0.7^7
2.P(X = 3) = 0.2668

AP Statistics: Unit 2 (topics 2.10 The Binomial Distribution). This calculator supports Unit 2 topic 2.10, The Binomial Distribution, where the Fall 2026 AP course expects you to justify a binomial setting and compute its probabilities, mean, and standard deviation.

What the binomial probability calculator does

A binomial distribution calculator works out the probability of getting a set number of successes in a fixed number of independent trials. You give it three inputs: the number of trials $n$ , the probability of success on each trial $p$ , and the count of successes $x$ you care about.

The tool returns the exact probability $P(X = x)$ , where $X$ (capital X) is the random variable that counts successes. It also handles range questions, like at least four successes or at most three successes, by adding the right individual probabilities.

You do not need the formula to use the calculator, but knowing what it computes helps you read the output and check your work by hand.

The binomial probability formula

The exact probability of $x$ successes comes from the binomial probability function on the AP formula sheet:

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

Here $\binom{n}{x}$ counts the ways to choose which $x$ of the $n$ trials are the successes, $p^x$ is the probability those trials succeed, and $(1-p)^{n-x}$ is the probability the rest fail. The values of $x$ run from $0$ to $n$ .

The calculator evaluates this for whatever $x$ you enter, so you can read a single-outcome probability straight off.

At least and at most: cumulative probability

Most real questions ask for a range, not a single value. A cumulative probability adds the individual probabilities across a range of counts.

For at most $k$ successes, add every probability from $0$ up to $k$ :

P(X \le k) = P(0) + P(1) + \dots + P(k)

For at least $k$ successes, it is usually faster to use the complement rule than to add a long tail:

P(X \ge k) = 1 - P(X \le k-1)

So at least four successes equals one minus the chance of three or fewer. The calculator lets you pick exactly, at most, at least, or a between range, and it applies the matching sum for you.

Mean and standard deviation

A binomial distribution has a simple center and spread. The mean (the symbol $\mu_X$ , read mu, is the average number of successes) is

\mu_X = np

The standard deviation (the symbol $\sigma_X$ , read sigma, measures the typical spread around the mean) is

\sigma_X = \sqrt{np(1-p)}

Both are on the AP formula sheet. Read them in context: $\mu_X$ is how many successes you expect on average, and $\sigma_X$ is how far actual counts tend to fall from that average.

How to use the calculator

Enter the number of trials $n$ .
Enter the probability of success $p$ as a decimal between 0 and 1.
Choose the type of question: exactly, at most, at least, or between.
Enter the success count, or the two endpoints for a between range.

The output shows the probability, plus the mean and standard deviation for the whole distribution. Check that your $p$ is a decimal, not a percent, since 30 percent is entered as 0.3.

Conditions checklist (BINS)

A setting is binomial only when all four conditions hold. A common memory aid is BINS:

Binary: each trial has two outcomes, a success or a failure.
Independent: the result of one trial does not change the others.
Number: the number of trials $n$ is fixed in advance.
Same probability: the success probability $p$ is the same on every trial.

If the number of trials is not fixed, or you are counting trials until the first success, the setting is geometric instead of binomial. See binomial vs geometric for how to tell them apart.

Binomial probability for n = 10, p = 0.3

A quiz has 10 independent multiple-choice questions, and a student guesses each one with a 0.3 chance of being correct. Let $X$ be the number of correct answers. Find $P(X = 3)$ , then $P(X \ge 4)$ , and give the mean and standard deviation of $X$ .

Check the setting is binomial: 10 fixed trials, two outcomes per question (right or wrong), independent guesses, and the same $p = 0.3$ each time. So $X$ is binomial with $n = 10$ and $p = 0.3$ .
Set up the exact probability with the formula: $P(X = 3) = \binom{10}{3} (0.3)^3 (0.7)^7$ .
Evaluate each piece: $\binom{10}{3} = 120$ , $0.3^3 = 0.027$ , and $0.7^7 = 0.0823543$ .
Multiply: $P(X = 3) = 120 \times 0.027 \times 0.0823543 = 0.2668$ .
For at least four, use the complement: $P(X \ge 4) = 1 - P(X \le 3)$ .
Add the lower tail: $P(0) = 0.0282$ , $P(1) = 0.1211$ , $P(2) = 0.2335$ , $P(3) = 0.2668$ , so $P(X \le 3) = 0.6496$ .
Subtract: $P(X \ge 4) = 1 - 0.6496 = 0.3504$ .
Mean: $\mu_X = np = 10 \times 0.3 = 3$ correct answers.
Standard deviation: $\sigma_X = \sqrt{np(1-p)} = \sqrt{10 \times 0.3 \times 0.7} = \sqrt{2.1} = 1.4491$ .

$P(X = 3) = 0.2668$ and $P(X \ge 4) = 0.3504$ . The distribution has mean $\mu_X = 3$ correct answers and standard deviation $\sigma_X \approx 1.4491$ .

Frequently asked questions

What is the difference between at least and at most on a binomial calculator?

At most $k$ means $k$ or fewer successes, so you add $P(0)$ through $P(k)$ . At least $k$ means $k$ or more, so you add $P(k)$ through $P(n)$ , or faster, take $1 - P(X \le k-1)$ . Watch the wording, because more than 4 is $P(X \ge 5)$ , not $P(X \ge 4)$ .

Do I enter p as a decimal or a percent?

Enter $p$ as a decimal between 0 and 1. A 30 percent success rate is 0.3, and a 5 percent rate is 0.05. Entering 30 instead of 0.3 gives output that makes no sense.

When can I not use the binomial distribution?

When any BINS condition fails. If the number of trials is not fixed, the trials are not independent, the probability of success changes, or there are more than two outcomes, the setting is not binomial. Counting trials until the first success is geometric, not binomial.