Calculate binomial probabilities: exact P(X = k), cumulative P(X ≤ k), and upper-tail P(X ≥ k) for any number of trials, successes, and success probability.
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0.18
P(X = 8)
0.6
P(X ≤ 8)
0.58
P(X ≥ 8)
| P(X = 8) — exact probability | 0.18 |
| P(X ≤ 8) — cumulative | 0.6 |
| P(X ≥ 8) — upper tail | 0.58 |
| P(X > 8) | 0.4 |
| Mean (n · p) | 8 |
| Variance (n · p · (1−p)) | 4.8 |
| Standard deviation | 2.19 |
Also in Statistics
Probability Distributions
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Enter the number of trials (n), number of successes (k), and probability of success (p) to get exact and cumulative probabilities.
The binomial distribution applies when: (1) there are a fixed number of trials n; (2) each trial has exactly two outcomes — success or failure; (3) the probability of success p is constant; and (4) trials are independent.
Classic examples: the number of heads in n coin flips, the number of defective items in a sample of n products, the number of patients who respond to treatment out of n patients.
P(X = k) is the exact probability of getting exactly k successes. P(X ≤ k) is the cumulative probability of getting k or fewer successes. P(X ≥ k) is the upper-tail probability of getting k or more successes.
For a significance test, the p-value is 2 × min(P(X ≤ k), P(X ≥ k)) for a two-tailed test, or the relevant one-tailed probability.
Frequently asked questions
P(X = k) is the exact (point) probability of getting exactly k successes. P(X ≤ k) is the cumulative probability of k or fewer successes — the sum of P(X = 0) through P(X = k). P(X ≤ k) is needed for one-tailed hypothesis tests.
When n is large and p is not close to 0 or 1, the normal approximation applies. A common rule: n·p ≥ 5 and n·(1−p) ≥ 5. For small n or extreme p, use exact binomial probabilities as computed here.
Yes. The calculator uses logarithmic arithmetic to compute binomial coefficients for large n without numerical overflow, handling n in the hundreds or thousands accurately.
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