What is the binomial distribution?

The binomial distribution models the number of successes k in n independent Bernoulli trials with success probability p. P(X=k) = C(n,k) × p^k × (1-p)^(n-k).

When do I use binomial vs normal?

Use binomial for exact counts. When np≥5 and n(1-p)≥5, the normal approximation works well. For large n, normal approximation simplifies calculations.

How do I compute P(X≥k)?

P(X≥k) = 1 - P(X≤k-1) = 1 - CDF(k-1). Or sum PMF from k to n. For "at least k" we use the complement of "at most k-1".

What are the mean and variance?

Mean E(X) = np. Variance Var(X) = np(1-p). Standard deviation σ = √(np(1-p)). These follow from the sum of n independent Bernoulli trials.

When are binomial trials valid?

Trials must be independent, each with the same success probability p, and we count the number of successes. Sampling with replacement (or large population) maintains independence.

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PROBABILITYDistributionsStatistics Calculator

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P(X=k), P(X≤k), P(X≥k) with PMF/CDF Charts

Probability tables. From coin flips to clinical trials. Binomial coefficient, expected value, variance, normal approximation hint.

Concept Fundamentals

C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ

Binomial PMF

Probability mass function

n trials, 2 outcomes

Requirements

Bernoulli trials

μ = n·p

Mean

Expected successes

σ² = n·p·(1−p)

Variance

Spread measure

Compute Probabilitiesn trials, p success probability

Why This Statistical Analysis Matters

Why: The binomial distribution models repeated independent trials with two outcomes. Used for coin flips, pass/fail tests, clinical trials, quality control.

How: Enter n (trials), p (success probability), and k or range. PMF = C(n,k)×p^k×(1-p)^(n-k). CDF = sum of PMF. Mean = np, Variance = np(1-p).

●np≥5 and n(1-p)≥5: normal approximation works
●CDF gives P(X≤k)
●At least k = 1 - P(X≤k-1)

Sources:NIST/SEMATECH e-HandbookKhan Academy - Binomial

B(n,p)

STATISTICSDistributions

P(X=k), P(X≤k), P(X≥k) with PMF/CDF Charts

Probability tables. From coin flips to clinical trials. Binomial coefficient, expected value, variance, normal approximation hint.

Normal Distribution →Poisson →

Real-World Scenarios — Click to Load

Calculation Mode

Inputs

Number of trials (n)

Probability of success (p)

Number of successes (k)

binomial.sh

CALCULATED

$ compute_binomial --n=10 --p=0.5 --k=5 --mode=exact

Primary Probability

24.6094%

E(X)

5.0000

Var(X)

2.5000

1.5811

Skewness

0.0000

Mode

Binomial Distribution

P(X=5)

24.6094%

n = 10p = 0.5E(X) = 5.00

numbervibe.com/calculators/statistics/binomial-distribution-calculator

PMF Bar Chart — P(X=k)

CDF Line Chart — P(X≤k)

Probability Table

k	P(X=k)	P(X≤k)	P(X≥k)
0	0.000977	0.000977	1.000000
1	0.009766	0.010742	0.999023
2	0.043945	0.054688	0.989258
3	0.117188	0.171875	0.945313
4	0.205078	0.376953	0.828125
5	0.246094	0.623047	0.623047
6	0.205078	0.828125	0.376953
7	0.117188	0.945313	0.171875
8	0.043945	0.989258	0.054688
9	0.009766	0.999023	0.010742
10	0.000977	1.000000	0.000977

Calculation Breakdown

INPUT

Parameters

n=10, p=0.5, k=5

COMPUTATION

Binomial Coefficient

C(10,5) = 252

n!/(k!(n-k)!)

RESULT

Primary Probability

24.6094%

P(X = 5) = \binom{10}{5} \cdot 0.5^{5} \cdot (1-0.5)^{10-5}

E(X)

5.0000

ext{np}

1.5811

√( ext{np}(1-p))

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

The binomial distribution models the number of successes in n independent trials with constant probability p
Must satisfy BINS: Binary outcome, Independent trials, Number of trials fixed, Same probability each trial
Mean = np, Variance = np(1-p), Standard Deviation = √(np(1-p))
When np ≥ 5 and n(1-p) ≥ 5, the normal approximation is valid
The binomial coefficient C(n,k) counts the number of ways to choose k successes from n trials

Did You Know?

🎲Jacob Bernoulli published the binomial distribution in Ars Conjectandi (1713) — 8 years after his deathSource: History of Math

📐Pascal's Triangle directly gives binomial coefficients — row n contains C(n,0), C(n,1), ..., C(n,n)Source: Combinatorics

🏭In manufacturing QC, the binomial distribution determines acceptable defect rates — a 2% defect rate in 100 items means P(≤3 defects) ≈ 85.7%Source: ISO

🧪Clinical trials use the binomial to test drug efficacy — comparing observed success rate against expected rateSource: FDA

💻A/B testing in tech uses the binomial to determine if a new feature improves conversion rates with statistical significanceSource: Tech

🗳️Election polling models are fundamentally binomial — each voter is a Bernoulli trial with probability p of supporting a candidateSource: Political Science

How It Works

1. Bernoulli Trials

Each trial has exactly two outcomes (success/failure), trials are independent, and the probability p is the same for every trial.

2. The Binomial Coefficient

C(n,k) counts the number of ways to arrange k successes among n trials — it comes from Pascal's triangle.

3. The PMF Formula

P(X=k) = C(n,k) p^k (1-p)^(n-k) — multiply the number of arrangements by the probability of each arrangement.

4. The CDF

The cumulative distribution function sums P(X≤k) — use it for "at most k" questions.

5. Normal Approximation

When np ≥ 5 and n(1-p) ≥ 5, use z = (k - np)/√(np(1-p)) with continuity correction (±0.5).

Expert Tips

Binomial vs Poisson

When n is large and p is small, Poisson with λ=np is simpler and accurate

Normal Approximation

When np ≥ 5 and n(1-p) ≥ 5, use normal with continuity correction (±0.5)

Check Independence

Sampling without replacement violates independence — use hypergeometric instead

One-Tailed vs Two-Tailed

For hypothesis tests, decide whether you need P(X ≥ k) or P(X ≤ k) before looking at data

Why Use This Calculator vs Other Tools?

Feature	This Calculator	Excel	R	Manual
PMF + CDF charts	✅	⚠️	⚠️	❌
Probability table	✅	❌	❌	⚠️
Multiple modes	✅	⚠️	⚠️	✅
Step-by-step	✅	❌	❌	✅
Normal approx hint	✅	❌	❌	⚠️
AI analysis	✅	❌	❌	❌

Frequently Asked Questions

What are the conditions for a binomial distribution (BINS)?

BINS: Binary outcome (success/failure), Independent trials, Number of trials fixed (n), Same probability (p) for each trial.

What is the difference between PMF and CDF?

PMF gives P(X=k) — the probability of exactly k successes. CDF gives P(X≤k) — the cumulative probability of at most k successes.

When can I approximate the binomial with a normal distribution?

When np ≥ 5 and n(1-p) ≥ 5. Use mean μ=np, σ=√(np(1-p)), and apply continuity correction (±0.5) when converting discrete to continuous.

What does "expected value" mean for a binomial distribution?

E(X) = np is the average number of successes you expect in n trials. For example, with n=100 and p=0.5, you expect 50 successes on average.

How is the binomial distribution related to Pascal's triangle?

Row n of Pascal's triangle contains C(n,0), C(n,1), ..., C(n,n) — the binomial coefficients used in the PMF formula.

Why is independence important for the binomial distribution?

If trials affect each other (e.g., sampling without replacement), the probability changes — use the hypergeometric distribution instead.

Binomial Distribution by the Numbers

C(n,k)

Pascal's Triangle

np ≥ 5

Normal Approx Rule

1713

Bernoulli Published

BINS

4 Conditions Required

Official Data Sources

NIST/SEMATECH e-Handbook ↗

US government statistical methods reference

Updated: 2025-12-01

Khan Academy - Binomial ↗

Free binomial distribution courses

Updated: 2026-01-15

Wolfram MathWorld ↗

Mathematical properties and proofs

Updated: 2025-11-20

OpenIntro Statistics ↗

Free open-source statistics textbook

Updated: 2024-06-01

SciPy binom Documentation ↗

Python implementation reference

Updated: 2026-01-28

Bernoulli - Ars Conjectandi ↗

Original 1713 work on binomial distribution

Updated: 1713-01-01

Disclaimer: This calculator provides binomial probabilities for educational and professional reference. For critical applications (clinical trials, quality control, financial modeling), verify results against established statistical software. The binomial model assumes BINS conditions; if trials are not independent, consider the hypergeometric distribution.

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⬅️Jump in and explore the concept!

P(X=k), P(X≤k), P(X≥k) with PMF/CDF Charts

Why This Statistical Analysis Matters

P(X=k), P(X≤k), P(X≥k) with PMF/CDF Charts

Real-World Scenarios — Click to Load

Calculation Mode

Inputs

PMF Bar Chart — P(X=k)

CDF Line Chart — P(X≤k)

Probability Table

Calculation Breakdown

Key Takeaways

Did You Know?

How It Works

1. Bernoulli Trials

2. The Binomial Coefficient

3. The PMF Formula

4. The CDF

5. Normal Approximation

Expert Tips

Binomial vs Poisson

Normal Approximation

Check Independence

One-Tailed vs Two-Tailed

Why Use This Calculator vs Other Tools?

Frequently Asked Questions

What are the conditions for a binomial distribution (BINS)?

What is the difference between PMF and CDF?

When can I approximate the binomial with a normal distribution?

What does "expected value" mean for a binomial distribution?

How is the binomial distribution related to Pascal's triangle?

Why is independence important for the binomial distribution?

Binomial Distribution by the Numbers

Official Data Sources

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Poisson Distribution Calculator

Normal Distribution Calculator

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