1.6.3 problem 58

Explanation: https://math.stackexchange.com/questions/1936525/inclusion-exclusion-problem

(a)

Let $A$ be the event that at least $9$ widgets need to be tested. $$P\left(A\right)=1-P\left(A^c\right)=1-\frac{\left(\frac{8}{3}\right)3!9!}{12!}$$

(b)

Similar to part $a$, $$P\left(A\right)=1-P\left(A^c\right)=1-\frac{\left(\frac{9}{3}\right)3!9!}{12!}$$