Probability Calculator Guide

What This Calculator Does

This probability calculator covers four common but non-interchangeable tasks: permutation counts, combination counts, exact-k binomial probabilities, and exact-k Poisson probabilities. The selector puts “how many arrangements or selections?” and “how likely is an exact event count?” in one place without pretending they are the same calculation.

Choose the matching mode when you need nPr, nCr, binomial P(X=k), or Poisson P(X=k). The binomial and Poisson modes also show expectation, variance, standard deviation, and coefficient of variation so you can compare a distribution’s center and relative spread. For P values, use the P Value Calculator; for standardized scores, use the Z-Score Calculator; for ordinary numeric expressions, use the Mathematical Expression Calculator.

When to Use It

• You need a permutation or combination count and know whether order matters.
• You have a fixed number of independent binary trials with success probability p and need exactly k successes.
• You model event counts with an average rate λ per interval and need exactly k events.
• You want expectation, variance, standard deviation, and relative variability alongside the headline result.
• You have already selected the model and need a quick, checkable numerical result.

Do not treat this page as a general probability solver for conditional probability, Bayes’ theorem, event unions/intersections, normal, geometric, or hypergeometric distributions, or at-least/at-most cumulative probabilities.

Inputs Explained

Permutation and Combination: n and r

Both modes require non-negative integer n and r, with r ≤ n.

• Permutation nPr: different orders count as different outcomes, such as assigning first, second, and third place.
• Combination nCr: only the selected members matter, such as choosing a committee.

The probability card in these modes shows 1 ÷ nPr or 1 ÷ nCr. Read it only as the reciprocal of the count; it is not a universal event probability based on a sample space you entered.

Binomial Distribution: n, k, and p

• n: number of independent trials, a non-negative integer.
• k: number of successes, a non-negative integer with k ≤ n.
• p: success probability on each trial, with 0 ≤ p ≤ 1.

The binomial model assumes fixed independent trials, two complementary outcomes per trial, and a constant success probability. This page calculates exactly k successes.

Poisson Distribution: λ and k

• λ: average number of events in a time or space interval, greater than 0.
• k: exact number of events in that interval, a non-negative integer.

λ is an average rate, not the probability of one event. The page does not decide whether a real data set satisfies the assumptions of a Poisson model.

How the Calculation Works

Permutation

P(n,r) = n! / (n-r)!

Combination

C(n,r) = n! / [r!(n-r)!]

Binomial Point Probability

If X is the number of successes in n trials:

P(X=k) = C(n,k) × p^k × (1-p)^(n-k)

The page also shows:

• Expected value: E(X) = n × p
• Variance: Var(X) = n × p × (1-p)
• Standard deviation: √Var(X)
• Coefficient of variation: standard deviation ÷ expected value (shown as 0 when the expected value is 0)

Poisson Point Probability

P(X=k) = (λ^k × e^-λ) / k!

It also shows:

• Expected value: E(X) = λ
• Variance: Var(X) = λ
• Standard deviation: √λ
• Coefficient of variation: 1 ÷ √λ

Example

Example 1: Permutation versus combination

Enter n=5 and r=3:

• Permutation: 5P3 = 5! / 2! = 60; the helper probability is 1/60 ≈ 0.01666667, or about 1.666667%.
• Combination: 5C3 = 5! / (3!×2!) = 10; the helper probability is 1/10 = 0.1, or 10%.

The counts differ because permutations include order and combinations do not. The helper probability is simply the reciprocal of each count.

Example 2: Exactly three binomial successes

Enter n=10, k=3, and p=0.5:

P(X=3) = C(10,3) × 0.5^3 × 0.5^7 = 0.1171875

The page also shows:

• Expected value: 10×0.5 = 5
• Variance: 10×0.5×0.5 = 2.5
• Standard deviation: √2.5 ≈ 1.58113883
• Coefficient of variation: 1.58113883 ÷ 5 ≈ 0.31622777

The 0.1171875 answers only “exactly three,” not “at least three” or “at most three.”

Example 3: Exactly two Poisson events

Enter λ=4 and k=2:

P(X=2) = (4^2 × e^-4) / 2! ≈ 0.146525

Because a Poisson distribution has mean and variance equal to λ, the page shows expected value 4, variance 4, standard deviation 2, and coefficient of variation 0.5.

How to Understand the Result

• Permutation and combination results are counts, not real-world probabilities by themselves.
• Binomial and Poisson headline results are point probabilities from 0 to 1; the card also displays a percentage.
• Expected value describes the long-run center of repeated trials or interval counts; it does not guarantee one observed result.
• Variance measures spread in squared units. Standard deviation uses the original unit and is usually easier to compare directly.
• Coefficient of variation expresses standard deviation relative to the expected value, which helps compare relative spread across different means.
• The page keeps up to eight decimal places and switches to scientific notation for very large or very small values. It is a formatted JavaScript-number result, not symbolic or arbitrary-precision arithmetic.

Common Mistakes

• Confusing permutations and combinations: ask whether order matters.
• Applying a binomial model when trials are not independent, the success probability changes, or there are more than two outcomes.
• Treating Poisson λ as a one-event probability instead of an average count per interval.
• Reading P(X=k) as a cumulative “at least” or “at most” probability.
• Entering negative or decimal n, r, or k, or using r/k > n.
• Expecting conditional probability, Bayes, normal, t-test, confidence-interval, or full distribution-table features.
• Using very large factorial inputs without accounting for JavaScript overflow and floating-point limits.

FAQ

What is the difference between nPr and nCr?

nPr counts arrangements, so different orders are different. nCr counts selections, so order is ignored. For the same n and r with r>1, the permutation count is usually larger.

Can the page calculate “at least k”?

No. The binomial and Poisson modes calculate only the exact-k point probability. At-least, at-most, and interval probabilities require summing multiple points or using a cumulative-distribution tool.

Can I enter p as a percentage?

Enter a decimal from 0 to 1—for example, enter 0.5 for 50%, not 50. The result card shows both decimal and percentage forms.

Can λ equal zero?

The current page requires λ to be greater than 0. A zero-rate Poisson model is a degenerate boundary case outside this component’s input range.

Why does permutation or combination mode show a probability field?

It is a helper reciprocal of the count. It does not use a sample space, equal-likelihood assumption, or event definition, so formal work should follow the probability definition in the problem.

Notes

This tool is useful for study, homework checking, and moderate-size discrete-probability calculations. It uses JavaScript Number values and direct factorial formulas; large factorials can overflow or accumulate floating-point error, so the displayed result is a formatted numerical approximation.

The page has four modes only: permutation, combination, binomial point probability, and Poisson point probability. It does not provide free-form probability modeling, conditional probability, Bayes, normal/geometric/hypergeometric distributions, cumulative distributions, simulation, confidence intervals, or statistical-inference conclusions. Confirm the model assumptions before relying on a result.