Binomial Distribution Calculator (CDF)

Calculate P(X ≤ k) for binomial trials.

Please enter a valid number of trials (1-500)

Examples

Enter n and p to generate the distribution

What Is Cumulative Binomial Probability?

While the Binomial Probability Calculator finds P(X = k)—the probability of exactly k successes—cumulative probabilities answer broader questions:

  • P(X ≤ k): At most k successes
  • P(X < k): Fewer than k successes
  • P(X ≥ k): At least k successes
  • P(X > k): More than k successes

These cumulative probabilities sum individual binomial probabilities.

The Formulas

P(X ≤ k) = Σ P(X = i) for i = 0 to k

P(X ≥ k) = 1 - P(X ≤ k-1) = Σ P(X = i) for i = k to n

P(X < k) = P(X ≤ k-1)

P(X > k) = 1 - P(X ≤ k)

Examples

Quality Control

"A process has 5% defect rate. In 50 items, what's P(at most 3 defectives)?"

n = 50, p = 0.05, find P(X ≤ 3)

P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.077 + 0.202 + 0.261 + 0.220 = 0.760

76% chance of 3 or fewer defectives.

Clinical Trials

"Drug works 80% of the time. If 20 patients take it, what's P(at least 15 respond)?"

n = 20, p = 0.80, find P(X ≥ 15)

P(X ≥ 15) = 1 - P(X ≤ 14) = 0.804

About 80% chance of 15+ responses.

Survey Analysis

"30% support a policy. In 100 respondents, what's P(fewer than 25 supporters)?"

n = 100, p = 0.30, find P(X < 25) = P(X ≤ 24)

P(X ≤ 24) ≈ 0.142

About 14% chance.

Game Theory

"Win probability is 40%. In 10 games, what's P(more than 6 wins)?"

n = 10, p = 0.40, find P(X > 6) = 1 - P(X ≤ 6)

P(X > 6) = 1 - 0.945 = 0.055

About 5.5% chance of winning 7 or more.

Applications

Hypothesis Testing for Proportions

One-sided test: "Is the success rate greater than 50%?"

Observe k successes in n trials. P-value = P(X ≥ k) under null hypothesis (p = 0.50)

This is an exact binomial test.

Acceptance Sampling

"Accept a batch if at most 2 of 50 tested items are defective."

P(accept good batch) = P(X ≤ 2 | p = 0.02) [producer's risk] P(accept bad batch) = P(X ≤ 2 | p = 0.10) [consumer's risk]

Control Charts

"Process is out of control if > 3 of 20 consecutive points are beyond 2σ."

Under control: P(point beyond 2σ) ≈ 0.05 P(false alarm) = P(X > 3 | n=20, p=0.05)

Sports Analytics

"Player's free throw percentage is 75%. What's the chance of making fewer than 15 of 25?"

P(X < 15 | n=25, p=0.75) helps assess underperformance probability.

Relationship to Exact Probabilities

Cumulative probabilities build from exact probabilities:

P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

Conversely, exact probabilities come from cumulative differences:

P(X = k) = P(X ≤ k) - P(X ≤ k-1)

Normal Approximation

For large n, the binomial approaches normal:

  • Mean μ = np
  • SD σ = √(np(1-p))

With continuity correction:

P(X ≤ k) ≈ Φ((k + 0.5 - np) / √(np(1-p)))

The +0.5 adjusts for the discrete-to-continuous conversion.

When to use: np ≥ 5 AND n(1-p) ≥ 5

The approximation is faster for very large n where exact calculation is slow.

Confidence Intervals and Sample Size

Cumulative binomial probabilities underlie:

  • Clopper-Pearson exact confidence intervals for proportions
  • Sample size calculations for proportion estimation
  • Power analysis for proportion tests

Computing Cumulative Probabilities

Direct Summation

Add individual terms P(X = 0) + P(X = 1) + ... + P(X = k)

Works well for small k.

Complement Method

For large k: P(X ≥ k) = 1 - P(X ≤ k-1)

Compute the smaller sum and subtract from 1.

Recursive Calculation

P(X = k+1) = P(X = k) × (n-k)/(k+1) × p/(1-p)

Start with P(X = 0) = (1-p)ⁿ and iterate.

Two-Sided Probabilities

For "extreme" events in either direction:

P(|X - np| ≥ c) = P(X ≤ np - c) + P(X ≥ np + c)

Used in two-sided hypothesis tests.

Relationship to Other Distributions

Poisson approximation: When n large, p small, np moderate: P(X = k) ≈ e^(-λ) × λ^k / k! where λ = np

Beta distribution: The cumulative binomial relates to the incomplete beta function, used in computing exact confidence intervals.

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Summary

Cumulative binomial probabilities answer practical questions about "at least," "at most," "fewer than," or "more than" a specified number of successes. From quality control acceptance decisions to hypothesis testing to risk assessment, these probabilities extend the basic binomial framework to real-world questions. For large n, the normal approximation with continuity correction provides a computationally efficient alternative.

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