Binomial Distribution Calculator

Binomial Distribution Formula

The probability mass function (PMF) for a binomial distribution is:

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Where:

• n = number of trials
• k = number of successes
• p = probability of success on a single trial
• C(n,k) = binomial coefficient = n! / (k! × (n-k)!)

Distribution Statistics

• Mean (μ) = n × p
• Variance (σ²) = n × p × (1-p)
• Standard Deviation (σ) = √(n × p × (1-p))

Cumulative Distribution Function (CDF)

P(X ≤ k) = Σ_i=0..k P(X = i)

When to Use Binomial Distribution

The binomial distribution applies when:

• There is a fixed number of trials (n)
• Each trial is independent
• Each trial has exactly two possible outcomes (success or failure)
• The probability of success (p) is the same for each trial