Binomial Distribution Calculator
Binomial Distribution Calculator Info
The Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.
Within sporting events a Binomial Distribution can be used to model things like the number of free throws made out of a number of attempts.
Formula
The probability mass function of a Binomial distribution is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials
- p is the probability of success in a single trial
- k is the number of successes
- (n choose k) is the binomial coefficient, the number of ways to choose k successes from n trials
Properties
- Mean: n*p
- Variance: n*p*(1-p)
- Standard Deviation: √(n*p*(1-p))
Normal Approximation
When the number of trials is sufficiently large, a Binomial distribution can be approximated by a normal distribution (the De Moivre–Laplace theorem). A common rule of thumb is that the approximation is reasonable when both n*p ≥ 5 and n*(1-p) ≥ 5 (and even better when both are ≥ 10). The approximating normal distribution uses:
- Mean (μ) = n*p
- Standard Deviation (σ) = √(n*p*(1-p))
Because the normal distribution is continuous while the binomial is discrete, a continuity correction of ±0.5 is applied to approximate discrete probabilities:
- P(X = k) is approximated by P(k - 0.5 ≤ X ≤ k + 0.5) (Between Probability in the Normal calculator)
- P(X ≤ k) is approximated by P(X ≤ k + 0.5) (Left Tail in the Normal calculator)
- P(X ≥ k) is approximated by P(X ≥ k - 0.5) (Right Tail in the Normal calculator)
The Related Calculators links below to the Normal Distribution calculator provide pre-filled links for P(X = k), P(X ≤ k), and P(X ≥ k) with the continuity correction applied.