Probability Mass Function Calculator

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.

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.

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.

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.

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.

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.

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.

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.