When is a situation binomial?
Check the four conditions (often remembered as BINS): Binary outcomes (success/failure), Independent trials, a fixed Number of trials, and the Same probability of success each time. Ten coin flips, 20 multiple-choice guesses, and defective items in a batch of 50 (if the defect rate is stable) all qualify.
Worked example
A student guesses all 10 questions on a quiz with 4 options each (p = 0.25). P(exactly 3 right) = C(10, 3) × 0.25³ × 0.75⁷ = 120 × 0.015625 × 0.1335 ≈ 0.2503. P(at least 6 right) ≈ 0.0197, so passing by guessing is unlikely.
“At least” problems are often easiest via the complement: P(X ≥ 1) = 1 − P(X = 0).
Frequently asked questions
What is the difference between binompdf and binomcdf?
binompdf gives P(X = k), a single bar. binomcdf gives P(X ≤ k), the running total of bars from 0 to k. For P(X ≥ k) use 1 − binomcdf(n, p, k − 1).
When can I use the normal approximation?
A common rule is when np ≥ 10 and n(1 − p) ≥ 10. This calculator computes the exact binomial, so you don’t need the approximation.