Question

Problem 8 A large box of fuses contains 10% defectives. Four fuses are randomly selected from the box. Find: a) Probability that exactly one fuse is defective b) Probability that at least one fuse of the four selected is defective Now suppose these four sampled fuses are shipped to a customer before being tested. Assume the cost of fixing' a shipment with defective fuses is given by C- 3Y2 where Y is the number of defectives in the shipment of four. Find the expected repair cost E(C)-E3Y

Answer 1

8)

Let Y denote the number of defective fuses. Then, Y~Bin(n=4,p=0.10)

$P(Y=y)=\binom{4}{y}0.1^{y}(1-0.1)^{4-y},y=0,1,2,3,4$

So, $E(Y)=np=4(0.1)=0.4,Var(Y)=np(1-p)=4(0.1)(1-0.1)=0.36$

a)

Required probability = P(Y=1) = $\binom{4}{1}0.1^{1}(1-0.1)^{4-1}=0.2916$

b)

Required probability = $P(Y\geq 1)=1-P(Y=0)=1-\binom{4}{0}0.1^{0}(1-0.1)^{4-0}=0.3439$

Now, $E(C)=E(3Y^{2})=3E(Y^{2})=3[Var(Y)+E(Y)^{2}]=3[0.36+0.4^{2}]=0.1728$