binomial distribution online calculator

What is a binomial distribution?

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes: success or failure. If you have a process like flipping a coin, checking whether an email is opened, or counting defect-free items from a production line, a binomial model is often the right fit.

• The number of trials is fixed: n
• Each trial is independent
• Each trial has two outcomes (success/failure)
• The probability of success stays constant: p

Binomial formula (PMF)

For exactly k successes out of n trials, the probability is:

P(X = k) = C(n, k) · p^k · (1 - p)^n-k

where C(n, k) is the number of combinations. This is also called the binomial probability mass function (PMF).

How to use this binomial distribution online calculator

1. Enter the number of trials n.
2. Enter the success probability p (between 0 and 1).
3. Choose the probability type: exact, at most, at least, or between.
4. Enter the success count(s) k.
5. Click Calculate to view probability, percentage, and key distribution stats.

Interpreting your results

Exact probability: P(X = k)

Use this when you need one specific outcome. Example: exactly 4 people convert out of 10 visitors.

Cumulative probability: P(X ≤ k)

Use this for “at most” questions. Example: probability of getting 3 or fewer defective units in a batch.

Upper-tail probability: P(X ≥ k)

Use this for threshold events. Example: probability that at least 8 out of 12 students pass.

Range probability: P(k₁ ≤ X ≤ k₂)

Use this when outcomes in an interval matter. Example: probability that between 45 and 55 customers purchase.

Real-world examples

• A/B testing: How likely is at least 30 conversions out of 100 visits if conversion rate is 25%?
• Quality control: What is the chance of exactly 2 defects in 20 items if defect probability is 5%?
• Healthcare: Probability that at most 1 adverse reaction occurs in 15 patients.
• Sports analytics: Chance a player makes exactly 7 shots in 12 attempts.

Mean, variance, and standard deviation

Every binomial distribution has useful summary metrics:

• Mean: μ = n·p
• Variance: σ² = n·p·(1-p)
• Standard deviation: σ = √(n·p·(1-p))

This calculator shows these values so you can quickly understand the center and spread of outcomes.

Common mistakes to avoid

• Using percentages like 65 instead of probabilities like 0.65.
• Forgetting that n and k must be integers.
• Applying a binomial model when trials are not independent.
• Using variable success probability across trials (that violates assumptions).

FAQ

Can I use this as a cumulative binomial calculator?

Yes. Choose “At most” for cumulative lower-tail probability and “At least” for upper-tail probability.

What if p = 0 or p = 1?

The calculator handles those edge cases automatically. If p=0, only 0 successes are possible. If p=1, all trials are successes.

Is this good for hypothesis testing?

It can help compute exact probabilities used in binomial tests, but full statistical inference may require additional test setup.