Binomial and Poisson distributions
binomial: P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, E = np, V = np(1 − p)
The two coursework discrete distributions: the binomial (k successes in n independent trials of probability p) and Poisson (k events when λ are expected on average). The tool gives the exact probability, both cumulative probabilities. Each summed from ITS OWN SIDE so the small tail keeps its significant digits, the mean and the standard deviation. Everything runs through logarithms of factorials (direct summation, no coefficient-based approximation): n = 10,000 never overflows.
0.246094
- Calculation
- P(X = k) = C(10, 5) × 0.5^5 × (1 − 0.5)^(10−5)
Probability 0.246094 of exactly 5 successes in 10 trials (mean 5). Both cumulatives are summed from their own side, the small tail keeps its digits.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasbinomial: P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, E = np, V = np(1 − p)
binomial: P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, E = np, V = np(1 − p)
Poisson: P(X = k) = e⁻λ · λᵏ / k!, E = V = λ
cumulatives: P(X ≤ k) = Σᵢ₌₀ᵏ P(X = i), P(X ≥ k) = Σᵢ₌ₖ P(X = i)
The binomial counts successes of independent trials with the same probability, the two conditions of the model. Poisson is its limit as n grows and p shrinks with np = λ held constant: the law of rare events (calls, failures, decays). Internally everything runs in log space (ln C(n,k) from sums of ln i, never a direct factorial) so C(10000, 5000) × p⁵⁰⁰⁰ neither overflows nor vanishes; each cumulative is summed from its own side, so that P(X ≥ 95) out of 100 trials keeps its digits instead of dying inside 1 − 0.9999….
- Independent trials
- · one trial’s outcome does not influence the others, and p is identical at every trial: drawing with replacement. Drawing without replacement follows the hypergeometric distribution, not the binomial.
- Mean and variance
- · E = np for the binomial: over 100 flips at p = 0.5 you expect 50 successes, yet P(X = 50) is only 8%: the most probable value is still improbable.
- λ, mean AND variance
- · the Poisson signature: mean and variance equal. Counts whose variance clearly exceeds the mean (overdispersion) do not follow Poisson.
Validity domainn is capped at 10,000 and λ at 10,000, beyond that.
n is capped at 10,000 and λ at 10,000, beyond that. The normal approximation (normal distribution tool) is excellent anyway and standard practice. The binomial assumes constant p and independent trials: correlated series and draws without replacement are outside the model. Probabilities smaller than ~10⁻³⁰⁰ fall below the floating-point floor and are returned as 0.
Common pitfall: “at least one” is not computed term by term“What is the probability of at least one success?
“What is the probability of at least one success?” goes through the complement: P(X ≥ 1) = 1 − P(X = 0) (a single term) never by hand-summing P(1) + P(2) + …. A surprising example: at p = 1/100 and n = 100 trials, P(at least one) = 1 − 0.99¹⁰⁰ ≈ 63%, not 100%, “a one-in-a-hundred chance. A hundred times” is not a certainty. The tool gives both cumulatives directly to cut these manual sums short.