Probability Distribution Calculator
Calculate probabilities for the three distributions you meet most often. Pick binomial, normal, or Poisson, enter the parameters, and read the probability at your point (PMF or PDF), the cumulative probability up to it (CDF), and the distribution mean and variance.
How to use the probability distribution calculator
- Choose a distribution: binomial, normal, or Poisson.
- Enter its parameters, such as n, k and p for the binomial.
- Read the point probability, the cumulative CDF, and the mean and variance.
Examples
Binomial: 5 heads in 10 fair coin flips
Binomial, n = 10, k = 5, p = 0.5
P(X = 5) ≈ 0.2461, P(X ≤ 5) ≈ 0.6230, mean 5, variance 2.5
Normal: z = 1.96 on the standard normal
Normal, x = 1.96, mean = 0, sd = 1
Density ≈ 0.0584, P(X ≤ 1.96) ≈ 0.9750
Poisson: 0 events when the average is 1
Poisson, k = 0, lambda = 1
P(X = 0) ≈ 0.3679, P(X ≤ 0) ≈ 0.3679, mean 1, variance 1
Frequently asked questions
What is the difference between the PMF, PDF, and CDF?
The PMF (probability mass function) gives P(X = k) for a discrete distribution like the binomial or Poisson. The PDF (probability density function) is its continuous counterpart for the normal distribution, where the height f(x) is a density rather than a probability. The CDF (cumulative distribution function) gives P(X <= k), the chance of getting that value or anything lower.
Which distribution should I use?
Use the binomial for a fixed number of independent yes/no trials with the same success probability, such as heads in 10 coin flips. Use the Poisson for counts of events over a time or area when you only know the average rate, such as calls per hour. Use the normal for continuous, bell-shaped data like heights or measurement errors, and as an approximation to the binomial or Poisson when the numbers are large.
How is the normal CDF calculated without a z-table?
The calculator converts your value to a z-score, z = (x - mean) / sd, then evaluates the standard normal CDF using the Abramowitz and Stegun error-function approximation (formula 7.1.26). Its absolute error is under 1.5e-7, so the results match a printed z-table to several decimal places, and you do not need to look anything up.
What do the mean and variance tell me?
The mean is the long-run average outcome and the variance measures spread, with the standard deviation being its square root. For the binomial the mean is n*p and the variance is n*p*(1 - p). For the Poisson the mean and the variance are both lambda. For the normal you supply the mean and standard deviation directly, so the variance is simply sd squared.
Can it handle large values without breaking?
Yes. Binomial coefficients and factorials grow far too fast to store directly, so the calculator works in logarithms using a log-gamma function and only converts back at the end. That keeps probabilities accurate for large n or lambda, where a naive factorial would overflow to infinity or lose precision.
Is my data private?
Yes. Every calculation runs entirely in your browser using JavaScript. Your numbers are never uploaded, logged, or sent to a server, so you can use the tool offline once the page has loaded and nothing you type leaves your device.
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