About

In statistical analysis and experimental design, determining the likelihood of a specific number of successes within a fixed set of independent trials is fundamental. This calculator solves the Binomial Probability Mass Function for scenarios where only two outcomes are possible: success or failure. It is essential for researchers verifying hypothesis tests, quality control engineers assessing defect rates, and students tackling combinatorics problems.

Accuracy in these calculations is critical when dealing with small sample sizes or rare events, where approximation methods like the Normal distribution fail. This tool computes the exact probability for a specific count of successes, alongside the cumulative probabilities necessary for "at least" or "at most" type questions often found in academic exams and risk assessment models.

Formulas

The probability of exactly k successes in n independent Bernoulli trials is given by the Binomial Probability formula:

P(X = k) = n!k!(n − k)! ⋅ p^k ⋅ (1 − p)^{n − k}

Where n! represents the factorial of the total trials. The cumulative probabilities are summations of this function:

P(X ≤ k) = k∑i=0 n!i!(n − i)! pⁱ(1 − p)^{n − i}

Reference Data

Parameters	Symbol	Description	Typical Range
Number of Trials	n	Total count of independent experiments	1 ≤ n ≤ 100
Success Probability	p	Likelihood of success in a single trial	0.0 to 1.0
Target Successes	k	Exact number of successes to calculate	0 ≤ k ≤ n
Failure Probability	q	Calculated as 1 − p	0.0 to 1.0
Probability Mass	P(X=k)	Exact chance of obtaining k successes	0 to 1
Cumulative Lower	P(X≤k)	Chance of obtaining k or fewer successes	0 to 1
Cumulative Upper	P(X≥k)	Chance of obtaining k or more successes	0 to 1
Mean	x	Expected value (n ⋅ p)	Depends on n
Variance	σ²	Spread of distribution (n ⋅ p ⋅ q)	Positive Real

Frequently Asked Questions

Use this distribution when your experiment meets three criteria: fixed number of trials (n), only two possible outcomes per trial (success/failure), and the probability of success (p) remains constant for every independent trial.

Factorials grow exponentially. For inputs larger than 170, standard calculators often return errors or Infinity. This tool is optimized to handle up to n=100 precisely, which covers most academic and standard quality control scenarios.

Cumulative probability helps answer broader questions. P(X ≤ k) tells you the chance of getting "at most" k successes. P(X ≥ k) tells you the chance of getting "at least" k successes. These are often more practically useful than the exact probability of a single specific number.

Probability distributions are easier to understand in context. By plotting k-1, k, and k+1, you can visually determine if your target value sits on the rising slope, the peak, or the falling tail of the distribution curve.

Bernoulli Distribution Calculator

Exact Probability P(X=k)

Cumulative P(X≤k)

Cumulative P(X≥k)

About

Formulas

Reference Data

Frequently Asked Questions