What This Calculator Does

Calculate permutation P(n,r), combination C(n,r), and binomial or Poisson point probabilities and cumulative tails. Choose exactly k, at most k or at least k. Distribution results also include expectation, variance, standard deviation and coefficient of variation.

When to Use It

Use for discrete-event counting and probability exercises with known assumptions. For normal-CDF percentiles use z-score; for a two-tailed mean test with known population standard deviation use P value. Conditional probability and Bayes remain outside this page.

Inputs Explained

Counting n is a whole number 0–170 and r is 0–n. Binomial n is 0–10000 and p a finite decimal 0–1. k is a whole number 0–100000. Poisson λ is finite and 0–10000. n is fixed trial count, p success probability per trial, and λ average count in a specified interval. Binomial k>n is a valid boundary: exact/at-least probability is zero and at-most is one.

How the Calculation Works

Binomial point probability is C(n,k)·p^k·(1−p)^(n−k), expectation np and variance np(1−p). Poisson point probability is e^−λ·λ^k/k!, with expectation and variance λ. At most k sums from zero through k; at least k sums from k upward. Both include k. P(X≥k) = 1−P(X≤k−1), not 1−P(X≤k). Log probabilities and compensated sums directly evaluate each tail, preserving small tails that could be lost by subtracting nearly equal numbers.

Example

Binomial n=10, p=0.5 and k=2: exact 45/1024 = 0.0439453125, at-most 56/1024 = 0.0546875, at-least 1013/1024 = 0.9892578125. Poisson λ=3 and k=2: exact about 0.22404181, at-most 0.42319008 and at-least 0.80085173. At-most and at-least do not necessarily sum to one because they overlap at k.