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🎯 Binomial Distribution Calculator

Calculate P(X = k), P(X ≤ k), and P(X ≥ k) for a binomial distribution. Enter the number of trials (n), probability of success (p), and number of successes (k) to get exact probabilities, the mean, variance, and a full distribution table.

Total number of trials (1–1000)

Probability per trial (0 to 1)

Specific number of successes to query

What is the Binomial Distribution?

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes (success or failure) and the probability of success is constant across all trials. These conditions are collectively known as a "Bernoulli process." Examples include: the number of heads in 10 coin flips, the number of defective items in a batch of 100, or the number of correct answers on a 20-question true/false test when guessing randomly.

The distribution is characterised by two parameters: n (the number of trials) and p (the probability of success on each trial). Its mean is μ = np and its variance is σ² = np(1−p). The probability of exactly k successes in n trials is given by the Probability Mass Function (PMF): P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) is the binomial coefficient "n choose k." The Cumulative Distribution Function (CDF) sums the PMF from 0 to k to give P(X ≤ k).

When n is large and p is close to 0.5, the binomial distribution approximates the normal distribution (by the Central Limit Theorem). When n is large and p is small (np < 5), it approximates the Poisson distribution. These approximations simplify calculations when working with very large n values where exact binomial computation becomes expensive.

Binomial Distribution Formulas

Probability Mass Function
P(X = k) = C(n, k) × p^k × (1−p)^(n−k)
Cumulative (at most k)
P(X ≤ k) = Σ P(X=i) for i = 0 to k
Mean, Variance, SD
μ = n × p
σ² = n × p × (1 − p)
σ = √(n × p × (1 − p))

C(n, k) is the number of ways to choose k successes from n trials. The binomial distribution models the number of successes in n independent trials, each with a constant success probability p.

How to Use This Calculator

  1. 1
    Enter n (trials)
    The total number of independent trials or experiments (e.g. 10 coin flips, 20 free throws).
  2. 2
    Enter p (probability)
    The probability of success on each single trial (0 to 1). For a fair coin, p = 0.5.
  3. 3
    Enter k (successes)
    The specific number of successes you want to query. The calculator gives P(X=k), P(X≤k), and P(X≥k).
  4. 4
    Read the table
    The distribution table shows the exact probability for every possible outcome from 0 to n, with your chosen k highlighted.

Worked Example

A basketball player has a free-throw success rate of p = 0.3. In n = 10 attempts, what is the probability of exactly k = 3 successes?

P(X=3) = C(10,3) × 0.3³ × 0.7⁷
= 120 × 0.027 × 0.0824
= 0.2668 (≈ 26.68%)
P(X≤3) = 0.6496
P(X≥3) = 0.6172
μ = 10 × 0.3 = 3.0, σ = √(10×0.3×0.7) = 1.449

How the Binomial Distribution Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

  1. Enter raw values or summary statistics.
  2. Clean separators and count the sample size.
  3. Apply the relevant statistic, probability, or confidence formula.
  4. Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

  • Samples should be representative of the population being studied.
  • Normality or independence assumptions apply only where the selected method requires them.
  • Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It answers questions like: If I flip a coin 20 times, what is the probability of exactly 8 heads?

Use it when: (1) there is a fixed number of trials n, (2) each trial has exactly two outcomes (success/failure), (3) the probability p is constant for every trial, and (4) trials are independent of each other. These are the BINS conditions.

The binomial is exact but discrete. The normal distribution is a continuous approximation that works well when both np ≥ 5 and n(1−p) ≥ 5. For small n or extreme p values, always use the exact binomial formula. For large n with moderate p, the normal approximation is accurate.

The expected value μ = n × p is the average number of successes you would expect over many repetitions of the experiment. For 10 coin flips with p = 0.5, you expect μ = 5 heads on average, though any single experiment may give a different result.

Real-World Applications

🏭
Quality Control
Manufacturing uses binomial distribution to model defect rates: "What is the probability that ≥2 items in a batch of 50 are defective if the defect rate is 3%?"
💊
Clinical Trials
Pharmaceutical research applies binomial probability to estimate success rates: "If a drug has 70% efficacy, what is the probability that at least 14 of 20 patients respond?"
📊
Survey Analysis
Polling uses binomial models to compute confidence intervals and determine whether observed results differ significantly from expected proportions.
🎲
Game Theory & Gambling
Calculating fair payouts, optimal strategies, and house edge in games with binary outcomes (win/lose) relies on binomial probability.
🌐
Network Reliability
Engineers model packet loss and node failures using binomial distributions to predict system uptime and design redundant architectures.
📈
Financial Risk
Binomial trees in options pricing (Cox-Ross-Rubinstein model) use repeated binomial steps to model asset price movements up or down.

Common Mistakes

1
Violating Independence Assumption
Binomial distribution requires trials to be independent. Drawing cards without replacement violates this — use the hypergeometric distribution instead.
2
Non-Constant Probability
If the probability of success changes between trials (e.g. fatigue reducing accuracy over time), the binomial model is invalid. Use more complex models for such scenarios.
3
Confusing PMF and CDF
P(X = k) gives the probability of exactly k successes (PMF). P(X ≤ k) gives the probability of at most k successes (CDF). Many real-world questions require the CDF.
4
Wrong n or p Input
Entering p = 70 instead of p = 0.70 (proportion vs percentage) is a common data-entry error that makes the result meaningless. Always verify units.
5
Using Normal Approximation Inappropriately
The normal approximation to the binomial requires np ≥ 5 and n(1−p) ≥ 5. For rare events (small p) or small n, use the exact binomial — do not approximate.

Discrete Distribution Comparison

Distribution Trials Use When
Binomial Fixed n, with replacement n fixed, p constant, independent trials
Hypergeometric Fixed n, without replacement Sampling from finite population without replacement
Poisson Events in time/space interval n large, p small, rare events (np < 5)
Negative Binomial Variable trials until k successes Counting trials needed to achieve k successes
Bernoulli Single trial (n=1) Single binary experiment

References

  1. Wackerly, D., Mendenhall, W. & Scheaffer, R. Mathematical Statistics with Applications. Cengage, 2008.
  2. Ross, S. M. Introduction to Probability Models. Academic Press, 2014.
  3. NIST/SEMATECH. e-Handbook of Statistical Methods — Binomial Distribution. nist.gov/sematech.
  4. DeGroot, M. H. & Schervish, M. J. Probability and Statistics, 4th ed. Pearson, 2012.
  5. Cox, D. R. & Rubinstein, M. Options Markets. Prentice-Hall, 1985.

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