A lot of 50 watches is known to have 10 defective watches. If 8 watches are selected one by one with a replacement at random, then the probability that there will be at least one defective watch is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(1-\left(\displaystyle\frac{4}{5}\right)^8\)
With replacement, the 8 draws are independent, each with
\( \displaystyle P(\text{defective}) = \frac{10}{50} = \frac{1}{5},\qquad P(\text{not defective}) = \frac{4}{5} \)
“At least one defective” is the complement of “no defective”:
\( \displaystyle P(\text{none defective}) = \left(\frac{4}{5}\right)^8 \)
\( \displaystyle P(\text{at least one defective}) = 1 - \left(\frac{4}{5}\right)^8 \)