Probability Calculator: Binomial & Poisson Cumulative Tails Guide

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.

How to Understand the Result

Probabilities appear as a 0–1 value and a percentage. Small values use scientific notation. Coefficient of variation is SD/expectation; when expectation is zero this page displays zero by convention, although relative variation is mathematically undefined. The counting helper probability is simply the reciprocal count, not a general real-world success rate.

Common Mistakes

Binomial trials must be independent with fixed p. Poisson requires a consistent time or space interval and stable average rate; do not use a daily λ for an hourly interval unchanged. Do not present an exact-k probability as an at-least answer.

FAQ

Does at least k include k?

Yes. P(X ≥ k) = 1 − P(X ≤ k−1), rather than 1 − P(X ≤ k).

What does the binomial model assume?

A fixed trial count, independent trials and the same success probability each time. Interpretation depends on those assumptions.

Are Bayes and conditional probability supported?

Not yet. This page offers counting, binomial and Poisson point probabilities and cumulative tails.

Notes

JavaScript floating-point results are approximate; sufficiently tiny probabilities underflow to zero. Poisson upper tails use scaled recurrence from the requested boundary or the mode, stopping at a relative-error bound. A distant k does not automatically zero a representable tail. Counts can exceed the safe-integer range and are approximate. No Bayes, conditional probability, simulations, fitting or formal inference. Inputs remain in the URL; old point-probability URLs default to exactly k.