Binomial probabilities

For X ~ B(n,p), the success probability p is constant, trials are independent and each has two outcomes. P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ for integer k from zero to n.

Translate the event carefully before using a calculator: at least one means 1 − P(X = 0), while more than three means X ≥ 4. A probability calculation does not establish that the binomial assumptions fit the real situation.

Worked example

For X ~ B(5,0.2), find P(X ≥ 1).

1. Use the complement: P(X ≥ 1) = 1 − P(X = 0).
2. P(X = 0) = 0.8⁵ = 0.32768.

Answer: 0.67232