Binomial Distribution Calculator

P(X = k) PMF & CDF Bar Chart Free · No Signup

Free online binomial distribution calculator: compute exact P(X = k), cumulative P(X ≤ k), or range P(a ≤ X ≤ b) probabilities for any number of trials n and success probability p. Interactive Plotly PMF chart, step-by-step KaTeX formulas, and Python scipy export.

Binomial Distribution Calculator
Number of independent Bernoulli trials
Probability on each trial (0 to 1)
Exactly k successes out of n trials

Result

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Enter values and click Calculate

Find binomial probabilities for your parameters.

Probability Mass Function

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Calculate to see the PMF bar chart.

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What Is the Binomial Distribution?

The binomial distribution models the number of successes in n independent Bernoulli trials, each with the same success probability p. It answers questions like “If I flip a coin 10 times, what’s the probability of getting exactly 6 heads?”

PMF:  P(X = k) = C(n, k) × pk × (1 − p)n−k
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Fixed Trials

The number of trials n is fixed in advance. Each trial is independent of the others.

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Two Outcomes

Each trial has exactly two outcomes: success (probability p) or failure (probability 1 − p).

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Constant Probability

The probability of success p remains the same for every trial.

Key Formulas & Statistics

Binomial Coefficient:  C(n, k) = n! / (k! × (n − k)!)
StatisticFormulaExample (n=10, p=0.5)
Mean (μ)n × p5.0
Variance (σ²)n × p × (1 − p)2.5
Std Dev (σ)√(n × p × (1 − p))1.5811
Skewness(1 − 2p) / σ0.0 (symmetric)
Worked Example: Flip a fair coin 10 times. What is P(X = 6)?
C(10, 6) = 210
P(X = 6) = 210 × 0.56 × 0.54 = 210 × 0.015625 × 0.0625 = 0.2051

Normal Approximation to Binomial

When n is large enough, the binomial distribution can be approximated by a normal distribution. This is useful for quick calculations without summing many PMF values.

Approximation:  X ~ N(np, np(1 − p))   when np ≥ 5 and n(1 − p) ≥ 5

Apply a continuity correction of ±0.5 for better accuracy. For example, P(X ≤ k) ≈ Φ((k + 0.5 − np) / √(np(1 − p))).

n = 30, p = 0.5

Real-World Applications

Quality Control

Number of defective items in a batch. If defect rate is 5%, what is the probability of finding 0–2 defects in 100 items?

Medicine & Clinical Trials

Number of patients responding to treatment. If a drug has 70% efficacy, what is P(at least 15 of 20 respond)?

Survey & Polling

Number of “yes” responses in a sample. If 30% of voters support a policy, how many in a sample of 50?

Sports & Games

Free throw success rates, coin tosses, dice outcomes. Classic probability scenarios modeled by binomial.

Frequently Asked Questions

A binomial distribution models the number of successes in n independent Bernoulli trials, each with the same success probability p. It is a discrete probability distribution where outcomes are counted as whole numbers from 0 to n.
Use the formula P(X = k) = C(n, k) × pk × (1 − p)n−k, where C(n, k) is the binomial coefficient “n choose k”. This calculator computes it automatically with step-by-step formulas.
PMF (probability mass function) gives the probability of exactly k successes: P(X = k). CDF (cumulative distribution function) gives the probability of at most k successes: P(X ≤ k), which is the sum of PMF values from 0 to k.
When np ≥ 5 and n(1 − p) ≥ 5, the binomial can be approximated by N(np, np(1−p)). Apply a continuity correction of ±0.5 for better accuracy.
Mean = n × p. Variance = n × p × (1 − p). Standard deviation = √(variance). Skewness = (1 − 2p) / √(np(1−p)). When p = 0.5, the distribution is symmetric.
Common examples include coin flips, quality control defect counts, survey yes/no responses, clinical trial outcomes, free throw success rates in basketball, and genetics where offspring inherit traits with fixed probability.

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