Binomial Distribution Calculator
This binomial distribution calculator gives the probability of exactly, at most or at least k successes in n trials, plus the mean, standard deviation and the full distribution table.
Example: Find: at most k successes: P(X ≤ k) · Number of trials n 10 · Probability of success p 0.5 · Number of successes k 3 → Probability: P(X ≤ 3) = 0.171875 (17.19%). Source: NIST/SEMATECH e-Handbook of Statistical Methods – Binomial Distribution. Updated: .
How it is calculated
In 10 coin flips, the chance of at most 3 heads is 17.19%. The binomial distribution describes how often an event with probability p occurs in n independent trials.
How it is calculated
For exactly k successes, P(X = k) = C(n, k) · pk · (1 − p)n − k. For “at most”, “at least” and “between”, the calculator adds the matching single probabilities directly instead of using 1 − P, so even very small probabilities stay accurate. Mean and variance are μ = n · p and σ² = n · p · (1 − p).
Good to know
- Conditions: each trial has only success or failure, p stays the same, and trials do not influence each other (drawing with replacement).
- Drawing without replacement from a small population calls for the hypergeometric distribution; for a large population the binomial is a good approximation.
- For large n and a middling p the shape resembles a normal distribution with μ = n · p and σ = √(n · p · (1 − p)).
- The table lists P(X = k) plus the cumulative P(X ≤ k) and P(X ≥ k), like the lookup tables in textbooks.
Frequently asked questions
How do you calculate a binomial distribution?
With P(X = k) = C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) is the binomial coefficient “n choose k”. For ranges, add the single probabilities.
What do at most and at least mean?
At most k means 0 to k successes, P(X ≤ k). At least k means k to n successes, P(X ≥ k). More than k means P(X > k) = P(X ≥ k + 1).
What are the mean and standard deviation?
The mean is μ = n · p and the standard deviation σ = √(n · p · (1 − p)). With 100 trials and p = 0.3, μ = 30 and σ ≈ 4.58.
When can you use the binomial distribution?
When there are n identical, independent trials with only two outcomes each and a constant success probability p, such as coin flips, a die (six or not) or drawing with replacement.
Sources and legal basis
- NIST/SEMATECH e-Handbook of Statistical Methods – Binomial Distribution
- Wikipedia – Binomialverteilung (Wahrscheinlichkeitsfunktion, Erwartungswert, Varianz, Modus)
- W. H. Press u. a.: Numerical Recipes – Gammafunktion (Lanczos-Näherung)
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