Calculate exact and cumulative binomial probabilities for n trials with success probability p. Explore the distribution with interactive charts and key statistics.
Enter the number of trials (n), probability of success (p), and select a probability query to explore the binomial distribution.
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes (success or failure) and the probability of success remains constant. It's one of the most important discrete probability distributions in statistics.
Where C(n, k) is the binomial coefficient "n choose k", representing the number of ways to choose k successes from n trials. This equals n! / (k! × (n-k)!).
μ = n × p
The average number of successes you expect over many repetitions.
σ² = n × p × (1-p)
Measures how spread out the distribution is from the mean.
σ = √(n × p × (1-p))
Square root of variance; easier to interpret as it's in the same units as k.
The binomial distribution is valid only when these conditions are met:
Exact Probability P(X = k): The probability of getting exactly k successes. Use this when you need to know the chance of a specific outcome.
Cumulative P(X ≤ k): The probability of getting k or fewer successes. Useful for questions like "at most" or "no more than."
Cumulative P(X ≥ k): The probability of getting k or more successes. Useful for questions like "at least" or "no fewer than."
Range P(a ≤ X ≤ b): The probability that the number of successes falls within a specific range.
Calculate probabilities under the normal curve
Convert between z-scores and probabilities
Compute various probability calculations
Build confidence intervals for means and proportions
Calculate mean, median, standard deviation, and more
Perform matrix calculations and linear algebra
Calculate probabilities for rare event occurrences
Apply linear transformations to random variables