How to use this calculator
Enter the number of trials n and the probability of success on each one, p (as a decimal: 20% is 0.2). Choose what to find and the number of successes k. The calculator adds the exact probabilities, so there is no rounding from a table or a normal approximation.
Formulas
Exactly k successesP(X = k) = C(n, k) pk (1 − p)n−k C(n, k) = n!k!(n − k)!
Mean and varianceμ = np σ² = np(1 − p)
Worked example
10 parts, each defective with probability 0.2: P(exactly 3 defective) (the default)
- C(10, 3) = 10!/(3! 7!) = 120.
- P(X = 3) = 120 × 0.2³ × 0.8⁷ = 120 × 0.008 × 0.2097 = 0.2013.
- For “at most 2”: P(0) + P(1) + P(2) = 0.1074 + 0.2684 + 0.3020 = 0.6778.
When is a situation binomial?
- A fixed number of trials n.
- Each trial is a success or a failure.
- The trials are independent, with the same probability p every time (sampling with replacement, or a small sample from a large batch).
Frequently asked questions
What is the difference between “at least 3” and “more than 3”?
At least 3 includes 3: P(X ≥ 3). More than 3 starts at 4: P(X > 3) = P(X ≥ 4). Mixing them up is the most common binomial mistake.
When can I use the normal approximation?
When np and n(1 − p) are both at least about 5. Use mean np, standard deviation √(np(1 − p)) and a continuity correction of 0.5.