Binomial Distribution Calculator
Approximate the binomial probability P(X = k) using the normal distribution. This method works well for large n and moderate p. For small n or extreme p, use exact binomial tables or statistical software.
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Formula
P(X = k) ≈ (1/√(2πnp(1−p))) × e^(−(k−np)²/(2np(1−p)))
This is the normal approximation to the binomial. The binomial mean is μ = np and the variance is σ² = np(1−p). We evaluate the normal probability density function at x = k with this mean and variance. The result approximates the probability of exactly k successes. The approximation improves as n increases. For best accuracy, n should be large enough that both np ≥ 5 and n(1−p) ≥ 5.
How to use the Binomial Distribution Calculator
- 1
Enter your number of trials (n)
- 2
Enter your number of successes (k)
- 3
Enter your success probability (%)
Value should be in %.
- 4
Read your results instantly
Results update in real time as you type.
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The binomial distribution and when it applies
The binomial distribution models the number of successes in n independent Bernoulli trials, where each trial has the same probability p of success. Classic examples: flipping a coin n times and counting heads, testing n products and counting defectives, giving n patients a treatment and counting recoveries.
For the exact binomial probability, you need: P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) = n!/(k!(n−k)!) is the binomial coefficient. This requires computing large factorials, which is impractical for large n without software.
The normal approximation and its accuracy
For large n, the binomial distribution is well-approximated by a normal distribution with the same mean (np) and variance (np(1−p)). This calculator uses the normal PDF evaluated at k to approximate P(X=k).
The approximation is most accurate when n is large and p is close to 0.5. It degrades when p is close to 0 or 1 (the distribution becomes skewed) or when n is small. A rule of thumb: the approximation is acceptable when both np ≥ 5 and n(1−p) ≥ 5. For n=10 and p=50%, both conditions give 5 — right at the edge.
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When to use exact vs. approximate methods
For small n (say, n ≤ 20) or extreme p (below 10% or above 90%), use exact binomial probability from a table or statistical software. Excel's BINOM.DIST function, R's dbinom(), and Python's scipy.stats.binom.pmf() all compute exact probabilities.
The normal approximation is historically important because it enabled probability calculations before computers. Today it is mainly used as a conceptual bridge between discrete and continuous distributions, and for approximate confidence intervals for proportions.
Tips & Insights
Check np ≥ 5 and n(1−p) ≥ 5
Before trusting the normal approximation, verify both conditions. If either is below 5, the approximation can be significantly off. Use exact binomial for small n or extreme p values.
Continuity correction improves accuracy
For better approximation of cumulative probabilities, apply a continuity correction: P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − np) / √(np(1−p))). Adding 0.5 accounts for the discrete-to-continuous conversion.
Mean and variance are np and np(1−p)
The expected number of successes is np. The variance is np(1−p). Standard deviation is √(np(1−p)). These are easy to compute and often all you need to describe the distribution.
Worked Examples
Quality control — defective items
Approx. P(X=8) ≈ 0.0530. There is about a 5.3% chance of exactly 8 defective items in a batch of 100 when the defect rate is 5%.
Coin flip — exactly 5 heads in 10 flips
Approx. P(X=5) ≈ 0.2523. The exact binomial probability is 0.2461, so the normal approximation is close but not perfect for n=10.
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Frequently Asked Questions
What is the binomial distribution?
The binomial distribution gives the probability of exactly k successes in n independent trials, each with success probability p. It applies when trials are independent and each has only two outcomes.
Why does this calculator use a normal approximation?
The exact binomial formula requires computing large factorials. The normal approximation is computationally simple and accurate for large n, making it practical for a formula-based calculator.
When is the normal approximation valid?
The approximation is valid when both np ≥ 5 and n(1−p) ≥ 5. Outside these bounds, use the exact binomial distribution from statistical software.
What is the mean and variance of a binomial distribution?
Mean = np (expected number of successes). Variance = np(1−p). Standard deviation = √(np(1−p)).
How does binomial relate to Poisson?
When n is large and p is small (rare events), the binomial distribution with parameters n and p is well approximated by a Poisson distribution with λ = np. The Poisson approximation is simpler since it only needs the single parameter λ.
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