What is the binomial distribution?

The binomial distribution models the number of successes you get when you repeat the same experiment n times. Each repetition is called a trial. Every trial has only two possible outcomes: success or failure. The probability of success is the same each time and is written as p. If those conditions are true, the total number of successes, X, follows a binomial distribution: X ~ Bin(n, p).

You see this distribution everywhere: “How many heads in 10 coin flips?” “How many customers click a button out of 200 impressions?” “How many parts pass inspection out of 50?” Even sports questions like “How many free throws will a player make out of 12 attempts?” can be modeled as binomial when the assumptions are reasonable. The power of the binomial model is that it gives you a clean way to calculate exact probabilities—no guesswork and no simulation required.

• Fixed number of trials (n): you decide the count in advance (10 flips, 50 inspections, 200 emails).
• Only two outcomes per trial: success/failure, pass/fail, click/no click, yes/no.
• Constant probability (p): the chance of success doesn’t change from trial to trial.
• Independence: trials don’t influence each other in a meaningful way.

• If p changes over time (learning effects, fatigue, seasonality), the binomial assumptions may be violated.
• If trials aren’t independent (sampling without replacement from a small population), consider a hypergeometric model.
• If you’re using a binomial approximation in real life, sanity-check with observed data and context.

The probability formula (PMF) step by step

The binomial probability mass function (PMF) tells you the probability of getting exactly k successes in n trials:

P(X = k) = C(n, k) · p^k · (1 − p)^{n − k}

Here’s what each piece means:

• k: the number of successes you care about (for example, exactly 4 heads).
• p^k: probability of success happening k times.
• (1−p)^n−k: probability of failure happening the remaining n−k times.
• C(n,k): “n choose k” = the number of different ways to arrange k successes among n trials.

The combination term is what makes the binomial powerful. Even if each specific sequence (like success-success-failure-success…) has the same probability, there are often many sequences that lead to the same total number of successes. The term C(n,k) counts those sequences.

The combination formula is: C(n,k) = n! / (k! · (n − k)!)

Factorials grow extremely fast, so calculators (including this one) typically compute combinations in a numerically stable way without writing huge factorials explicitly. That’s why you can safely use large n values here without overflowing your browser.

• P(X = k): exact probability for one value.
• P(X ≤ k): cumulative probability from 0 up to k (this is the CDF at k).
• P(X ≥ k): right tail probability—getting k or more successes.
• P(a ≤ X ≤ b): probability of landing in a whole interval.

What this calculator computes behind the scenes

When you click “Calculate Probability,” the calculator first validates your inputs: n must be a whole number ≥ 0, p must be between 0 and 1, and your k/a/b values must be integers within the range 0…n. Then it computes binomial probabilities using the PMF formula.

For exact probabilities, it calculates P(X=k) directly. For cumulative probabilities, it sums the PMF values: P(X≤k) = Σ P(X=i) for i=0…k. For the right tail it uses a stable identity: P(X≥k) = 1 − P(X≤k−1). That avoids summing a potentially long tail in the wrong direction. For a range, it sums the PMF from a…b.

Along with the probability, the calculator also shows the key summary statistics that most classes and textbooks expect: mean (n·p), variance (n·p·(1−p)), and standard deviation (the square root of the variance). These help you understand the “center” and “spread” of the distribution even if you don’t compute every probability.

Some events are naturally very unlikely. For example, if p=0.1 and n=50, getting 30 successes is extremely rare. That doesn’t mean the calculator is “wrong”—it means the event is far into the tail. If your answer prints as 0.000000…, use the Scientific Notation Calculator linked below to present it cleanly (or increase decimals).

Real examples you can copy

Example 1: Coin flips (exact).
Flip a fair coin n=10 times. What is the probability of getting exactly k=6 heads? Here p=0.5. The calculator computes P(X=6) using the PMF. You’ll get a value around 0.205 (about 20.5%), meaning “6 heads” is fairly common in 10 flips.

Example 2: Click-through rate (cumulative).
Suppose a button has a 3% click rate (p=0.03). You show it to n=100 visitors. What’s the probability of getting at most k=2 clicks? Choose mode P(X ≤ k), enter n=100, p=0.03, k=2. This returns a cumulative probability that tells you how often you’ll see 0, 1, or 2 clicks in 100 views.

Example 3: Quality control (right tail).
A factory produces parts with a 2% defect rate (p=0.02). If you sample n=50 parts, what is the probability of finding at least k=3 defective parts? Select P(X ≥ k) and enter n=50, p=0.02, k=3. This is a tail probability—great for “risk” style questions.

Example 4: Range probability.
A student answers n=20 true/false questions by guessing (p=0.5). What’s the probability they get between a=8 and b=12 correct (inclusive)? Select the range mode, enter n=20, p=0.5, a=8, b=12. This is a classic “middle band” probability question.

Many exam questions hide the binomial setup in words. Translate the story into (n, p, k) carefully: n = number of attempts, p = per-attempt success probability, k = number of successes you’re counting. Then identify whether the question is asking for exact, at most, at least, or a range.

Frequently Asked Questions

• What does “Bin(n, p)” mean?

  It means a binomial distribution with n trials and success probability p. The random variable X counts how many successes occur out of those n trials.

• Do n and k have to be integers?

  Yes. “Number of trials” and “number of successes” are counts, so they must be whole numbers. The calculator will ask you to fix them if you enter decimals.

• What if I need P(X < k) or P(X > k)?

  Use the provided modes and adjust by one: P(X < k) = P(X ≤ k−1) and P(X > k) = P(X ≥ k+1). In other words, shift the boundary to match “≤” or “≥”.

• Why does P(X ≥ k) use 1 − P(X ≤ k−1)?

  It’s numerically stable and faster. For a right tail, directly summing P(X=k)+P(X=k+1)+… can be longer and may lose precision for tiny probabilities. Subtracting a left CDF from 1 is often cleaner.

• What’s the difference between probability and expected value?

  Probability answers “how likely is this event?” Expected value (mean) answers “what’s the average number of successes over many repeats?” A mean of 3 doesn’t mean 3 will happen—it means 3 is the long-run average.

• Can I use this for A/B testing or conversion rates?

  Yes—binomial is the natural model for conversions (success = convert, failure = not convert) when each user view is treated as an independent trial with constant p. In practice, real systems may violate independence, but binomial is still the starting point for many analyses.

• Does the calculator use a normal approximation?

  No. This tool computes binomial probabilities by summing the exact PMF, so it works well even for small n. If you want approximations (normal/Poisson), we can add those later as optional modes.

Common mistakes (and how to avoid them)

• Mixing up p and (1−p): p is the probability of success. (1−p) is failure. Keep the definition consistent.
• Using the wrong mode: “Exactly” uses P(X=k). “At most” uses P(X≤k). “At least” uses P(X≥k).
• Forgetting inclusivity: P(a ≤ X ≤ b) includes both endpoints. If you need strict bounds, shift them.
• Invalid k: if k > n, the probability is 0. This calculator will flag the input instead of silently returning 0.
• Non-binomial scenario: if p changes each trial or trials aren’t independent, your answer may not match reality.

If you’re using this for homework, write your final statement in words: “The probability of getting at least 3 defects in 50 parts is …” That’s often worth points.