What conditions must be met for binomial probability?

There must be a fixed number of trials n, each trial is independent, each trial has exactly two outcomes (success/failure), and p is constant across all trials.

What is the expected value of a binomial distribution?

The expected value (mean) is μ = n × p. The standard deviation is σ = √(n × p × (1−p)).

When should I use normal approximation instead?

When n is large (typically n > 30) and both np and n(1−p) are greater than 5, the normal distribution provides a good approximation to the binomial.

Can p be greater than 1?

No. Probability must be between 0 and 1 inclusive. A p of 0 means the event never occurs; p of 1 means it always occurs.

What is the difference between binomial and Bernoulli distributions?

A Bernoulli distribution is a single trial (n=1). A binomial distribution is the sum of n independent Bernoulli trials.

Math

↵ to calculate

Binomial Probability Calculator - Distribution Formula

Calculate binomial probabilities P(X=k), P(X≤k), and P(X≥k) for a given number of trials, successes, and probability.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

The binomial probability calculator computes the probability of exactly k successes in n independent trials, where each trial has a fixed probability p of success. It also calculates cumulative probabilities: P(X ≤ k) and P(X ≥ k). This is used in statistics, quality control, medical trials, and game theory.

Trials (n)

Successes (k)

Probability (p)

What is Binomial Probability Calculator - Distribution Formula?

How to use it

1️⃣ Enter n — the total number of trials.
2️⃣ Enter k — the number of successes you want to find the probability for.
3️⃣ Enter p — the probability of success on each trial (between 0 and 1).
4️⃣ P(X=k), P(X≤k), and P(X≥k) are calculated instantly.

Formula

P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, where C(n,k) = n! / (k! × (n−k)!). P(X≤k) = Σᵢ₌₀ᵏ P(X=i). P(X≥k) = 1 − P(X≤k−1).

💡 See it in action — a real example

A coin is flipped 10 times (p=0.5). P(X=4): C(10,4) × 0.5⁴ × 0.5⁶ = 210 × 0.0625 × 0.015625 ≈ 0.2051 (20.51%). P(X≤4) ≈ 37.7%. P(X≥4) ≈ 82.8%.

❓ Common questions

What conditions must be met for binomial probability?: There must be a fixed number of trials n, each trial is independent, each trial has exactly two outcomes (success/failure), and p is constant across all trials.
What is the expected value of a binomial distribution?: The expected value (mean) is μ = n × p. The standard deviation is σ = √(n × p × (1−p)).
When should I use normal approximation instead?: When n is large (typically n > 30) and both np and n(1−p) are greater than 5, the normal distribution provides a good approximation to the binomial.
Can p be greater than 1?: No. Probability must be between 0 and 1 inclusive. A p of 0 means the event never occurs; p of 1 means it always occurs.
What is the difference between binomial and Bernoulli distributions?: A Bernoulli distribution is a single trial (n=1). A binomial distribution is the sum of n independent Bernoulli trials.

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