Problems this can answer
It models quality samples, repeated shots, responses, and yes-or-no trials under stable conditions. A related application of Binomial Distribution is first success.
Math calculator
Binomial Distribution Calculator
Find exact or cumulative success probability for independent Bernoulli trials. Each submitted value produces binomial probability plus the intermediate reasoning.
Binomial Distribution inputs
Start with the given values
Seeing the method in action
For n=12,p=.3, exactly four successes has probability about .2311. Cumulative modes add exact counts.
The idea behind the result
A binomial variable counts successes in n independent trials with constant probability p. Exactly k has probability C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. Binomial Distribution can also be compared with rth success.
The calculation treats probability type, trial count n, success count k and success probability p according to their labels, not as interchangeable values in binomial distribution.
Checking the Binomial Distribution result
Trial count is fixed, outcomes are binary, trials are independent, and p is constant. Violating a condition changes the model.
An impossible binomial probability sign or magnitude usually points to field assignment before it points to rounding.
Reproducing the answer
Choose the k successful positions, multiply success and failure factors, and sum exact probabilities for a requested tail.
Using the result beyond this page
Start the binomial distribution setup by pairing every source number with probability type, trial count n, success count k and success probability p. Convert unlike units before typing, and postpone rounding until the displayed binomial probability is ready to report.
Raising p shifts mass toward larger counts; increasing n changes both center np and spread. Use that direction of change to check the displayed binomial probability before copying it elsewhere.
When copying the result elsewhere, include its label and any squared, linear, angular, or percentage unit implied by the inputs. That record distinguishes a calculated binomial probability from an unlabeled number and makes later checking substantially easier.
Geometric distribution asks when the first success occurs instead of counting a fixed run. Keep that boundary in mind when interpreting the numerical result.
For later verification, record probability type, trial count n, success count k and success probability p before rounding the binomial probability. The unrounded working value can feed subsequent steps while the rounded value serves presentation.
Boundary cases for Binomial Distribution
Test Binomial Distribution first with modest values for Probability type, Trial count n, Success count k, Success probability p that can be evaluated without software. A tree, short table, Pascal row, or handwritten outcome list supplies a structurally different comparison and makes a misplaced factorial or complement visible.
For Binomial Distribution, next inspect the allowable endpoints. Probabilities must stay between zero and one, counts cannot become negative, and a requested subset cannot exceed its parent. Passing those checks supports the setup; it does not by itself prove that the real situation satisfies the model assumptions.
Questions about Binomial Distribution
What is the mean?
np.
What is the variance?
np(1−p).
Can p change?
Not in this model.
Does at least include k?
Yes.
Must trials be independent?
Yes.