Probability Mass Function Calculator

Calculate the Probability Mass Function (PMF) for the Binomial Distribution by entering the probability of success, number of successes, and number of trials. You get the exact probability P(X = k) along with the cumulative probability P(X ≤ k) and a chart showing the full binomial distribution. Also try the Hypergeometric Distribution Calculator.

Enter a value between 0 and 1 (e.g. 0.3 for 30% chance of success)

The exact number of successful outcomes you want to evaluate

Total number of independent trials in the experiment

Results

P(X = k) — Exact Probability

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P(X ≤ k) — Cumulative Probability

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P(X > k) — Complement Probability

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Distribution Mean (μ = np)

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Standard Deviation (σ)

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Results Table

Frequently Asked Questions

What is the Probability Mass Function (PMF)?

The PMF gives the probability that a discrete random variable takes on an exact value. For the binomial distribution, P(X = k) tells you the probability of getting exactly k successes in n independent trials, each with probability p of success. See also our calculate Negative Binomial Distribution P(X = x).

What is the formula for the binomial PMF?

The binomial PMF is P(X = k) = C(n, k) × p^k × (1 - p)^(n - k), where C(n, k) is the binomial coefficient 'n choose k', p is the probability of success, k is the number of successes, and n is the total number of trials.

What is the difference between PMF and CDF?

The PMF (Probability Mass Function) gives the probability of an exact outcome, P(X = k). The CDF (Cumulative Distribution Function) gives the probability of an outcome up to and including k, P(X ≤ k). The CDF is the sum of all PMF values from 0 to k.

When can I use the binomial distribution?

The binomial distribution applies when: (1) there are a fixed number of trials n, (2) each trial has only two outcomes (success or failure), (3) the probability of success p is the same for every trial, and (4) all trials are independent of one another. You might also find our use the Discrete Uniform Distribution Calculator useful.

What are the mean and standard deviation of the binomial distribution?

For a binomial distribution with n trials and success probability p, the mean is μ = np and the standard deviation is σ = √(np(1 - p)). These describe the center and spread of the distribution respectively.

Can the probability of success be 0 or 1?

Technically yes, but edge cases apply. If p = 0, then P(X = 0) = 1 and all other outcomes have probability 0. If p = 1, then P(X = n) = 1. For meaningful probabilistic analysis, p should be strictly between 0 and 1.

What does the complement probability P(X > k) mean?

P(X > k) is the probability of getting more than k successes in n trials. It equals 1 − P(X ≤ k). This is useful when you want to know the likelihood of exceeding a certain threshold of successes.

How accurate is the binomial PMF calculator for large n?

For large values of n, the binomial coefficients become very large and floating-point precision can be a concern. This calculator uses logarithmic computation of binomial coefficients to maintain accuracy even for large n values. For very large n (e.g. > 1000), the normal approximation to the binomial may also be considered.