MyCalcBuddy

Binomial Probability Calculator

Calculate exact and cumulative probabilities for binomial distributions

About Binomial Probability

The binomial probability formula calculates the probability of getting exactly k successes in n independent trials, where each trial has probability p of success.

Formula: P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Example: What is the probability of getting exactly 3 heads in 10 coin flips?

What Is Binomial Probability?

Binomial probability answers one of the most practical questions in statistics: given n independent attempts, each with probability p of success, what are the chances of achieving exactly k successes? The binomial framework models real-world scenarios from coin flips and quality inspections to clinical trial outcomes and marketing conversion rates.

The power of this calculator goes beyond the standard exact probability P(X=k). It computes all five key probability values in one view: exactly k, less than k, at most k, greater than k, and at least k. This comprehensive output saves you from manually summing probabilities and lets you see the full picture of your binomial experiment at a glance.

Alongside the probabilities, the calculator provides the mean np, variance np(1−p), standard deviation, and the number of combinations — everything needed to understand both the specific outcome and the overall distribution behavior.

The Binomial Probability Formula

The binomial probability formula combines combinatorial counting with the laws of independent probability. The binomial coefficient C(n,k) counts how many different ways k successes can be arranged among n trials, and each such arrangement has probability p^k × (1−p)^(n−k).

Binomial Probability Mass Function

P(X = k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) = n! / (k! × (n−k)!)

Where:

  • n= Total number of independent trials or experiments
  • k= Exact number of successes you want the probability for
  • p= Probability of success on a single trial (value between 0 and 1)
  • C(n,k)= Number of combinations — distinct ways to select k items from n

Interpreting All Five Probabilities

This calculator provides a complete set of probability perspectives for your binomial experiment. Understanding each one helps you answer different types of questions:

Probability Question Answered Example
P(X = k)Chance of exactly k successes?Exactly 3 heads in 10 flips?
P(X < k)Chance of fewer than k successes?Fewer than 3 defects in a batch?
P(X ≤ k)Chance of at most k successes?At most 2 failures in 20 tests?
P(X > k)Chance of more than k successes?More than 8 correct answers?
P(X ≥ k)Chance of at least k successes?At least 5 respondents say yes?

The cumulative probabilities (less than, at most, greater than, at least) are computed by summing individual exact probabilities. For large n, this summation can involve many terms — the calculator handles the computation efficiently. The mean np gives the expected number of successes, while the standard deviation tells you how much variability to expect around that mean.

How to Use This Calculator

Get started with three simple inputs:

  1. Number of Trials (n): Enter the total count of independent experiments. Each trial must be independent and have the same success probability. For example, n=10 for ten coin flips or n=100 for a batch of 100 products.
  2. Number of Successes (k): The specific number of successes you want to evaluate. Must be an integer between 0 and n. The calculator will compute all five probability views relative to this k value.
  3. Probability of Success (p): Enter a decimal between 0 and 1. 0.5 for a fair coin, 0.02 for a 2% defect rate, 0.75 for a 75% response rate. The calculator accepts any valid probability.
  4. Review all results: The output panel shows every probability format, plus distribution statistics and the combination count — giving you complete insight into the binomial experiment.

Real-World Applications

Binomial probability calculations drive decisions in acceptance sampling and quality control. A manufacturer drawing a sample of n items from a production lot uses P(X>k) to determine the risk of accepting a bad batch. If the defect rate is truly unacceptable, the probability of observing k or fewer defects should be very small — making acceptance unlikely.

In medicine and epidemiology, binomial probability quantifies the chance of observing a specific number of treatment responses or adverse events. When planning clinical trials, researchers compute P(X≥k) to estimate study power — the probability of detecting a treatment effect of a given magnitude with a specific sample size.

In survey sampling and market research, the binomial model helps interpret poll results. Given a sample of n respondents, the probability that exactly k favor a candidate lets pollsters compute margins of error and confidence intervals. In sports analytics, binomial probability models a player's free-throw shooting, a bowler's strike rate, or a batter's hit probability over a season.

Worked Examples

Survey Response Analysis

Problem:

A survey reaches 50 people. Historically, 60% of recipients respond. What is the probability that exactly 30 people respond? What is the probability that more than 35 respond?

Solution Steps:

  1. 1Step 1: Enter n = 50, k = 30, p = 0.60. This gives P(X=30) — the exact probability of 30 responses.
  2. 2Step 2: P(X=30) = C(50,30) × 0.60^30 × 0.40^20. The calculator handles the large binomial coefficient automatically.
  3. 3Step 3: P(X>30) = 1−P(X≤30). Set k = 30 and read the 'greater than' output. To check P(X>35), change k to 35.
  4. 4Step 4: The mean response count is np = 50 × 0.60 = 30. The variance = 50×0.60×0.40 = 12, standard deviation ≈ 3.46.

Result:

With mean 30 and standard deviation 3.46, the most likely response count is around 30. P(X>35) is relatively small — 35 is about 1.45 standard deviations above the mean, placing it roughly in the upper 7-8% tail of the distribution.

Defect Rate Investigation

Problem:

A production process claims a 1% defect rate. In a batch of 200 items, 5 are found defective. What is the probability of observing 5 or more defects if the claimed rate is correct? Does this raise suspicion?

Solution Steps:

  1. 1Step 1: Enter n = 200, k = 4, p = 0.01. We use k=4 because P(X≥5) = 1−P(X≤4).
  2. 2Step 2: The calculator computes P(X≤4) = sum of probabilities for 0 through 4 defects with p = 0.01.
  3. 3Step 3: P(X≥5) = 1−P(X≤4). This is the 'greater equal' probability — the chance of seeing a result as extreme as 5+ defects.
  4. 4Step 4: If P(X≥5) is very small (e.g., below 0.05), the observed 5 defects are unlikely under the claimed rate — warranting investigation.

Result:

With n=200 and p=0.01, the expected number of defects is only 2. Finding 5 defects has a relatively low probability of occurrence — approximately 5-6%. While not impossible, this result is near the 5% threshold often used in hypothesis testing, suggesting the process may warrant a closer look.

Free Throw Reliability

Problem:

A basketball player has an 80% free throw success rate. In a game, she takes 10 free throws. What is the probability she makes fewer than 7? At least 9?

Solution Steps:

  1. 1Step 1: Enter n = 10, k = 6, p = 0.80. P(X<7) = P(X≤6) appears in the 'at most' output.
  2. 2Step 2: For 'at least 9', set k = 9 and read P(X≥9) = P(X=9) + P(X=10).
  3. 3Step 3: P(X=9) = C(10,9)×0.80^9×0.20^1 = 10×0.1342×0.2 ≈ 0.268. P(X=10) = 0.80^10 ≈ 0.107.
  4. 4Step 4: P(X≥9) ≈ 0.268+0.107 = 0.375, or 37.5%. P(X<7) = P(X≤6) is the complement of making 7+.

Result:

She has a 37.5% chance of making at least 9 out of 10 free throws — a very strong performance. The probability of making fewer than 7 is about 12%, reflecting her high skill level: most games she'll make 7 or more.

Tips & Best Practices

  • ✓When computing P(X≥k), use k−1 for the 'less than' complement: P(X≥k) = 1−P(X≤k−1).
  • ✓The most likely value of k is near np — check whether your observed k is within one standard deviation of the mean.
  • ✓For hypothesis testing, set p to the null hypothesis value and compute P(X≥k_observed) for the p-value.
  • ✓With large n, the binomial coefficient C(n,k) can be astronomically large — the calculator handles this efficiently.
  • ✓Use P(X≤k) for acceptance sampling: accept a batch if the observed defect count is at most the acceptance number.

Related Calculators

Binomial Distribution Calculator

Standard binomial calculator with PMF and CDF

Probability Calculator

General probability computation and rules

Geometric Distribution Calculator

Count trials until the first success occurs

Binomial Test Calculator

Exact binomial test for proportion hypotheses

Frequently Asked Questions

P(X<k) is the probability of fewer than k successes — it sums probabilities for 0 through k−1. P(X≤k) adds the exact probability P(X=k) on top, covering 0 through k. The difference between them is exactly P(X=k). This distinction matters most when k is small — for k=0, P(X<0)=0 but P(X≤0)=P(X=0), which can be substantial when p is low.
While both compute binomial probabilities, this calculator provides P(X<k) and P(X≥k) in addition to the standard P(X=k), P(X≤k), and P(X>k). The five-probability output gives you a complete picture without needing to recalculate. It is particularly useful for hypothesis testing scenarios where 'at least' and 'fewer than' probabilities are frequently needed.
The binomial distribution becomes approximately normal when both np ≥ 10 and n(1−p) ≥ 10. This is a direct consequence of the Central Limit Theorem. When these conditions hold, you can use normal approximations with mean np and variance np(1−p), but this calculator always uses exact computation regardless of n.
As n increases, the probability mass spreads across more possible values of k. Even the most probable outcome (the mode, near np) can have a relatively low probability in absolute terms — for example, with n=100 and p=0.5, P(X=50) ≈ 0.08. This is because there are 101 possible outcomes sharing the total probability of 1.
Yes, the cumulative probabilities correspond directly to p-values. For a right-tailed test, the p-value is P(X≥k_observed) under the null hypothesis value of p. For a left-tailed test, use P(X≤k_observed). For a two-tailed test, double the smaller of the two one-tailed p-values. Always specify your hypothesis before computing the p-value.

Sources & References

Last updated: 2026-06-06

💡

Help us improve!

How would you rate the Binomial Probability Calculator?

<>

Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Standard Mathematical References

by Various

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.