An auto store has two suppliers for a given part. The store orders from Supplier A 70% of the time, and supplier B 30% of the time. Supplier A's parts are defective with probability 0.02, and supplier B's parts are defective with probability 0.05.
a. If I buy a part, what is the probability that it is defective?
b. I buy a part and it is defective. What is the probability that it comes from Supplier A?
c. Let's assume that the parts arrive from the factory in boxes of 10; if I pick two parts from the same box then I know they are from the same factory (but I don't know which factory they are from). If I buy two parts from the same box, what is the probability that they are both defective? Hint: What is the probability that two parts from supplier A are both defective? How about two parts from supplier B?
Okay everyone, here's what I think the answers are: for a, I multiply the defective probability by the percent I order from each supplier so I get
a. $0.7 \times 0.02 + 0.3 \times 0.05 = 0.029$
b. $(0.7 \times 0.02 + 0.3 \times 0.05) / 0.7$ . I know I need to find what percentage of defectiveness comes from Supplier A, I'm not sure I'm calculating this correctly. Any suggestions would be greatly appreciated.
c. $(0.7 \times 0.02 + 0.3 \times 0.05 ) (0.7 \times 0.02 + 0.3 \times 0.05 ) = 0.000841$
a. Your calculation is correct.
b. $$ Pr(\text{from A}|\text{defective}) = \frac{Pr(\text{defective}|\text{from A})Pr(\text{from A})}{Pr(\text{defective})} = \frac{0.02(0.7)}{0.029} = \frac{14}{29} = 0.4827586 $$
c.
$ Pr(\text{two parts are defective}|\text{the box is from A}) = 0.02^2 $
$ Pr(\text{two parts are defective}|\text{the box is from B}) = 0.05^2 $
$$ \begin{align} &Pr(\text{two parts are defective}) \\ &= Pr(\text{two parts are defective}|\text{the box is from A})Pr(\text{the box is from A}) + Pr(\text{two parts are defective}|\text{the box is from B})Pr(\text{the box is from B}) \\ &= 0.02^2(0.7) + 0.05^2(0.3) \\ &= 0.00103 \end{align} $$