Defective items in a batch

Problem

On a production line $2\%$ of the items are defective. A batch of $50$ items is inspected. Compute the probability of finding exactly $1$ defective item and that of finding at least $1$.

Solution

Each item is defective with probability $p=0,02$, independently; over $n=50$ items the binomial scheme holds.

Exactly one defective: $P_{50}(1) = \binom{50}{1}\,0,02\,(0,98)^{49} = 50\cdot 0,02\cdot (0,98)^{49} \approx \boxed{0,371}$

At least one defective (via the complementary event «no defective»): $P(k\ge 1) = 1 - (0,98)^{50} \approx \boxed{0,636}$

Links

Topics: Probabilita
Concepts: Distribuzione binomiale
Methods: Binomiale · Complementare prob
Skills: Calcolo probabilita
Exercise type: Problema probabilita