What is the binomial distribution used for?

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's used in quality control, clinical trials, survey sampling, and any scenario involving repeated yes/no experiments.

How do you calculate binomial probability?

Binomial probability uses the formula P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) is the binomial coefficient, n is the number of trials, k is the number of successes, and p is the success probability. The calculator automates this computation.

When should I use the binomial distribution?

Use the binomial distribution when you have a fixed number of independent trials, each trial has exactly two outcomes (success/failure), the probability of success stays constant across trials, and you want to count the total number of successes.

What's the difference between PMF and CDF?

The PMF (Probability Mass Function) gives the exact probability of a specific outcome, like P(X = 5). The CDF (Cumulative Distribution Function) gives the probability of getting that value or less, like P(X ≤ 5), by summing all PMF values up to that point.

What are the mean and variance of a binomial distribution?

The mean of a binomial distribution equals np (trials times success probability). The variance equals np(1-p). The standard deviation is the square root of the variance: √(np(1-p)). These formulas let you quickly assess the distribution's center and spread.

Binomial Distribution Explorer

Modify Parameters and See Results

Number of successes in n independent trials

Parameters

Number of Trials (n): 10

Success Probability (p): 0.50

Number of Trials (n)

The total number of independent trials or experiments

Success Probability (p)

0.50

The probability of success on each individual trial

Failure Probability (q = 1-p)

0.50

The probability of failure on each trial (complement of p)

Statistics

Expected Value

5.0000

Variance

2.5000

Std Deviation

1.5811

Mode

Probability Calculator

Enter value k:

Key Properties

Real-World Applications

Number of heads in 10 coin flips
Number of defective items in a batch of products
Number of students passing an exam (with fixed pass rate)
Number of successful free throws in basketball
Survey responses with yes/no questions

Using the Parameter Controls

Interpreting the PMF Chart

Understanding the CDF Graph

The Explorer at Its Opening Parameters

Calculating Point Probabilities

Computing Cumulative Probabilities

Range Probability Calculations

What is the Binomial Distribution?

Understanding Distribution Parameters

Normal Approximation Rule

Related Distributions and Tools

Using the Parameter Controls

Adjust n (number of trials) using the top slider to set how many independent experiments you're running. The slider ranges from 1 to 50 trials, allowing you to model scenarios from a single coin flip to extensive quality control batches.

Set p (success probability) with the bottom slider to define the likelihood of success on each trial. Values range from 0.01 to 0.99, with 0.5 representing equally likely outcomes like fair coin flips.

Watch the visualization update in real-time as you modify parameters. The blue bars show the probability mass function (PMF), displaying the exact probability for each possible number of successes from 0 to n.

Interpreting the PMF Chart

The PMF (Probability Mass Function) chart displays vertical bars for each possible outcome. The height of each bar represents P(X = k), the probability of getting exactly k successes.

The distribution's shape changes dramatically with different parameters. When p = 0.5, the distribution is symmetric. As p moves toward 0 or 1, the distribution becomes increasingly skewed.

The tallest bar indicates the mode - the most likely number of successes. For n trials with probability p, the mode is approximately at np, which matches the distribution's mean.

Understanding the CDF Graph

The CDF (Cumulative Distribution Function) shows P(X ≤ k), the probability of getting k or fewer successes. Unlike the PMF's discrete bars, the CDF displays as a step function that increases from 0 to 1.

Each step in the CDF corresponds to adding one more outcome's probability. The steeper the step, the higher the probability of that specific outcome.

Use the CDF to quickly find probabilities for ranges. For example, P(X ≤ 7) can be read directly from the CDF curve at x = 7.

The Explorer at Its Opening Parameters

The binomial view opens at

n = 10

and

p = 0.5

, giving eleven bars from

k = 0

k = 10

that sum to exactly

1

. The red line marks

E[X] = np = 5

, and the tallest bar sits there too, at

P(X = 5) = 0.2461

.

Switching to the CDF tab replots the same numbers as a running total that climbs from

0.001

1

Binomial, n = 10, p = 0.5: PMF above, CDF below

Eleven bars summing to exactly 1, symmetric about the peak at k = 5 where P = 0.2461 = 252/1024. Mean and mode coincide only because p = 0.5; the CDF below climbs from 0.001 to 1.

Mean and mode coincide here only because

p = 0.5

makes the coefficient row palindromic. Move

p

away from

0.5

and the mode becomes

\lfloor (n+1)p \rfloor

, which parts company with

np

.

The variance is

np(1-p) = 2.5

, so

\sigma \approx 1.58

. Almost the whole distribution lies within two standard deviations of the mean, which is why the bars at

k = 0

and

k = 10

—

1/1024

each — are barely off the axis.

Reading the two tabs together is the point of having both. The tallest PMF bar is the steepest CDF rise; a flat stretch in the CDF is a negligible PMF bar.

Calculating Point Probabilities

Enter a value k in the Point Probability calculator to find P(X = k), the exact probability of getting k successes. The calculator uses the binomial formula: C(n,k) × p^k × (1-p)^(n-k).

The binomial coefficient C(n,k) counts the number of ways to arrange k successes among n trials. The calculator handles this computation automatically, even for large values where manual calculation would be impractical.

Try calculating P(X = 5) for n = 10, p = 0.5. You should get approximately 0.246, meaning there's about a 24.6% chance of exactly 5 successes in 10 fair coin flips.

Computing Cumulative Probabilities

The Cumulative Probability calculator computes P(X ≤ k) by summing probabilities from 0 to k. This answers questions like "What's the probability of at most k successes?"

For the complement, use P(X > k) to find the probability of more than k successes. This equals 1 - P(X ≤ k) and is useful for threshold problems.

P(X ≥ k) includes k itself in the count, while P(X > k) starts counting from k+1. Understanding this distinction prevents off-by-one errors in probability calculations.

Range Probability Calculations

The Range Probability section calculates P(a ≤ X ≤ b), the probability that X falls between two values inclusive. Enter lower bound a and upper bound b to see how much probability mass lies in that interval.

Four range options handle boundary conditions differently:
• [a, b] includes both endpoints
• (a, b) excludes both endpoints
• [a, b) includes a, excludes b
• (a, b] excludes a, includes b

For discrete distributions, these distinctions matter. P(3 ≤ X ≤ 7) includes outcomes 3, 4, 5, 6, and 7, while P(3 < X < 7) includes only 4, 5, and 6.

What is the Binomial Distribution?

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's one of the most fundamental discrete probability distributions.

The distribution requires four conditions: (1) fixed number of trials n, (2) each trial has two outcomes (success/failure), (3) constant success probability p, and (4) trials are independent.

Common applications include quality control (number of defective items), clinical trials (number of patients responding to treatment), and survey sampling (number of favorable responses). For comprehensive theory and formulas, see binomial distribution theory page.

Understanding Distribution Parameters

The distribution has two parameters: n determines the number of trials, while p sets the success probability for each trial. Together, they completely specify the distribution's behavior.

The mean (expected value) equals np - if you have 100 trials with p = 0.3, expect about 30 successes on average. The variance equals np(1-p), measuring spread around the mean.

The standard deviation

\sqrt{np(1-p)}

indicates typical deviation from the mean. When p = 0.5, variance is maximized at n/4, creating the widest spread for a given n.

Normal Approximation Rule

When n is large and p isn't too close to 0 or 1, the binomial distribution approximates a normal distribution. The rule of thumb: use normal approximation when both np ≥ 5 and n(1-p) ≥ 5.

The approximating normal distribution has mean μ = np and standard deviation σ =

\sqrt{np(1-p)}

. Apply a continuity correction by adding/subtracting 0.5 to discrete values for better accuracy.

This approximation simplifies calculations for large n where computing exact binomial probabilities becomes computationally expensive. For detailed guidance on when to use this approximation, see normal approximation to binomial.