How to use the Binomial, Poisson and Normal Calculator
- 1. Pick the distributionBinomial for n trials, Poisson for counts at rate λ, normal for continuous data.
- 2. Enter parametersn and p, or λ, or μ and σ, then the value or range you need.
- 3. Read the probabilitySee the answer, mean, variance and a table of probabilities.
The three formulas
Binomial: P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, mean np, variance np(1 − p). Use it for a fixed number of independent yes/no trials.
Poisson: P(X = k) = e⁻ˡ λᵏ / k!, with mean = variance = λ. Use it for counts of rare events in a fixed time or space, such as defects per sheet.
Normal: z = (x − μ)/σ and P(X < x) = Φ(z), read from the standard normal table. Use it for measurements like heights or marks.
Worked examples
Binomial: a fair coin is tossed 10 times. P(exactly 3 heads) = C(10, 3)(0.5)¹⁰ = 120/1024 = 0.1172.
Poisson: a machine makes on average 2 defects an hour. P(no defect in an hour) = e⁻² = 0.1353, and P(at least one) = 1 − 0.1353 = 0.8647.
Normal: marks have μ = 50 and σ = 10. P(X < 60) has z = 1, so Φ(1) = 0.8413. P(60 < X < 70) = Φ(2) − Φ(1) = 0.9772 − 0.8413 = 0.1359.
Which one to use
If n is large and p is small with np moderate, the binomial is close to a Poisson with λ = np. If np and n(1 − p) are both above about 5, it is close to a normal with μ = np and σ = √(np(1 − p)). For more on one distribution, see the binomial probability calculator and the normal distribution calculator, which also draws the curve.
Frequently asked questions
When should I use Poisson instead of binomial?
Use Poisson when you count events over time or space with no fixed number of trials, or as an approximation when n is large and p is small.
Is P(X < x) the same as P(X ≤ x) for the normal distribution?
Yes. For a continuous variable the probability of any single exact value is 0.
How accurate is the normal probability?
The tool uses a numerical approximation of Φ(z) accurate to about seven decimal places, more than any printed table.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.