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.
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.
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.
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.
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.
Yes. P(X ≥ k) = 1 − P(X ≤ k−1), rather than 1 − P(X ≤ k).
A fixed trial count, independent trials and the same success probability each time. Interpretation depends on those assumptions.
Not yet. This page offers counting, binomial and Poisson point probabilities and cumulative tails.
Choose the model and distinguish exactly k, at most k and at least k.