Binomial Calculator — Step-by-Step Solutions

Q: When should I use the binomial distribution?

Use it when you have a fixed number of independent trials, each with exactly two outcomes (success/failure) and the same probability of success.

Q: What is the difference between exact and cumulative probability?

Exact probability P(X = k) is the chance of getting exactly k successes. Cumulative probability P(X <= k) is the chance of getting k or fewer successes, computed by summing all exact probabilities from 0 to k.

Q: Can n be very large?

Yes, but computation may slow down for extremely large n. For large n, the normal approximation to the binomial (with continuity correction) is often used instead.

Q: What happens when p = 0 or p = 1?

When p = 0, every trial fails, so P(X = 0) = 1. When p = 1, every trial succeeds, so P(X = n) = 1. All other probabilities are 0.

The Binomial Calculator computes probabilities for binomial experiments, where each trial has exactly two outcomes (success or failure). It handles exact probabilities, cumulative probabilities, and expected value calculations.

Enter the number of trials (n), probability of success (p), and number of successes (k) to get instant results. The probability distribution bar chart shows the likelihood of each possible outcome, giving you a clear visual picture of the distribution.

Trials (n)

Probability (p)

Successes (k)

Your calculations will appear here

How to Use the Binomial Calculator

Choose a mode: "Exact P(X=k)" calculates the probability of exactly k successes, "Cumulative (<=k)" sums probabilities from 0 to k, "Cumulative (>=k)" sums from k to n, and "Expected Value" gives the mean, variance, and standard deviation.
Enter the number of trials (n) and the probability of success on each trial (p, between 0 and 1). For modes other than Expected Value, also enter the number of successes (k).
Results appear instantly, including a probability distribution bar chart that highlights the value of interest.
Review the step-by-step working to understand the binomial formula application.

Formula and Theory

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It requires two parameters: n (the number of trials) and p (the probability of success on each trial).

The probability mass function is P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where C(n, k) = n! / (k!(n - k)!) is the binomial coefficient. This formula counts the number of ways to choose which k trials are successes, multiplied by the probability of that specific sequence of outcomes.

The expected value (mean) of a binomial distribution is E(X) = n * p, the variance is Var(X) = n * p * (1 - p), and the standard deviation is the square root of the variance. As n increases and p is not too close to 0 or 1, the binomial distribution approximates a normal distribution.

Worked Examples

Example 1: Exact probability

Problem: A coin is flipped 10 times. What is the probability of getting exactly 7 heads?

Solution: n = 10, k = 7, p = 0.5. P(X = 7) = C(10,7) * 0.5^7 * 0.5^3 = 120 * 0.0078125 * 0.125 = 0.1172.

Answer: P(X = 7) = 0.1172 (about 11.72%)

Example 2: Cumulative probability

Problem: A multiple choice test has 20 questions with 4 choices each. What is the probability of getting 8 or fewer correct by guessing?

Solution: n = 20, p = 0.25, k = 8. Sum P(X = 0) + P(X = 1) + ... + P(X = 8) using the binomial formula.

Answer: P(X <= 8) = 0.9591 (about 95.91%)

Example 3: Expected value

Problem: A fair die is rolled 60 times. What is the expected number of sixes?

Solution: n = 60, p = 1/6. E(X) = 60 * (1/6) = 10. Var(X) = 60 * (1/6) * (5/6) = 8.333.

Answer: Expected value = 10, standard deviation = 2.887

Frequently Asked Questions

When should I use the binomial distribution?

Use it when you have a fixed number of independent trials, each with exactly two outcomes (success/failure) and the same probability of success.

What is the difference between exact and cumulative probability?

Exact probability P(X = k) is the chance of getting exactly k successes. Cumulative probability P(X <= k) is the chance of getting k or fewer successes, computed by summing all exact probabilities from 0 to k.

Can n be very large?

Yes, but computation may slow down for extremely large n. For large n, the normal approximation to the binomial (with continuity correction) is often used instead.

What happens when p = 0 or p = 1?

When p = 0, every trial fails, so P(X = 0) = 1. When p = 1, every trial succeeds, so P(X = n) = 1. All other probabilities are 0.