Two defective pieces out of eight

Text

A machine produces pieces, each defective with probability $0.1$, independently of the others. A sample of $8$ pieces is taken. What is the probability that exactly $2$ are defective?

Solution

The number $X$ of defective pieces out of $8$ independent trials with “success” (defect) probability $p=0.1$ follows a binomial distribution $X\sim B(8;\,0.1)$: $P(X=k) = \binom{8}{k}(0.1)^{k}(0.9)^{8-k}.$ For $k=2$: $P(X=2) = \binom{8}{2}(0.1)^{2}(0.9)^{6} = 28\cdot 0.01\cdot 0.531441 \approx \boxed{0.149}.$