In the course of producing music CDs, 13% of CDs produced are defective Pr(detec
ID: 1136179 • Letter: I
Question
In the course of producing music CDs, 13% of CDs produced are defective Pr(detective) Pr(not defective) 13% -87% Suppose CDs can be tested to determine if they are defective, but the testing procedure is not perfect. In particular, 98% of nondefective CDs will receive a favorable rating, while only 11% of defective ones will. The following conditional probabilities describe the accuracy of the testing procedure Pr(favorable Pr(unfavorable Pr(favorable Pr(unfavorable I I 1 I not defective)-98% not defective)=2% defective) detective)-89% 11% a) Calculate the Pr(favorable and not defective)? b) Use similar calculations to fill out the following joint distribution table Joint Distribution Favorable Unfavorable Not Defective Defective c) What is Pr(favorable) and Pr(unfavorable)? d) The firm loses $1.50 on each defective CD sold. The firm makes S2.00 on each good CD sold (These are profits, net of production costs). What is the expected profit from selling a CDExplanation / Answer
a) P(Not defective) = 0.87
P(favourable if defective) = 0.98
Therefore, P( favourable and non defective) =P(Not defective) * P(favourable if defective) = 0.98*0.87 = 0.8526
b) Similar to a) we calculate the other probabilities as folows:
P( Favourable and defective) = 0.11 *0.13 = 0.0143
P( Unfavourable and not defective) = 0.02*0.87 = 0.0174
P( unfavourable and defective) = 0.89* 0.13 = 0.1157
The table is given as follows:
c) Pr (favourable) = Pr(Favourable and not defective) + Pr(favourable and defective) = 0.8526 + 0.0143 =0.8669
Pr(unfavourable) = Pr(unfavourable and not defective) + Pr(unfavourable and defective) = 0.0174+0.1154 = 0.1328
d) Pr(non defective) = Pr(favourable and non defective) + Pr(non favourable and and non defective) = 0.8526 + 0.0174 = 0.87
Pr(defective) = Pr(favourable and defective) + Pr(non favourable and defective) = 0.0143 + 0.1157 = 0.13
Expected profit = profit on good CD * Pr(non defective) - loss on defective CD * Pr(defective) = 2*0.87 - 0.13*1.5 = 0.195 + 1.74 = $1.935
Joint Distribution Favourable Unfavourable Not defective 0.8526 0.0174 Defective 0.0143 0.1157