How to read the table: each cell shows P(X = k) = C(n,k)·p^k·(1−p)^(n−k). Rows are the values of k (0, 1, …, n); columns are the values of p. Use the n selector to change the number of trials and the button to change the range of p.
Click a cell to read the value.
P(X = k) — p from 0.05 to 0.50
How to read the table: each cell shows P(X ≤ k) = Σᵢ₌₀ᵏ C(n,i)·p^i·(1−p)^(n−i). To calculate P(X > k) use the complement: 1 − P(X ≤ k).
Click a cell to read the value.
P(X ≤ k) — p from 0.05 to 0.50
How to use these tables
Binomial Distribution
A binomial experiment consists of n independent trials, each with probability p of success. The number of successes X follows a B(n, p) distribution with probability mass function:
\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n \)
The mean is μ = np and the variance σ² = np(1 − p).
Useful properties
- Symmetry: P(X = k; n, p) = P(X = n − k; n, 1 − p). Use this for p > 0.5 by looking at the equivalent column with p' = 1 − p.
- Complement: P(X > k) = 1 − P(X ≤ k).
- Interval: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1).
- For large n and small p, the binomial is well approximated by a Poisson distribution with λ = np.
- For large n, the binomial is well approximated by a Normal distribution N(np, np(1−p)).