Answer to Question #259272 in Statistics and Probability for Mayy

Question

Of the feed material for a manufacturing plant, 85% is satisfactory, and the rest not. If it is satisfactory, the probabifity it will pass Test A is 92%. If it is not satisfactory, the probability it will pass Test A is 9.5%. If it passes Test A it goes on to Test B; 99% will pass Test B if the material is satisfactory, and 16% will pass Test B if the material is not satisfactory. If it fails Test A it goes on to Test C; 82% will pass Test C if the material is satisfactory, but only 3% will pass Test Cif the material is not satisfactory. Material is accepted if it passes both Test A and Test B. Material is rejected if it fails both Test A and Test C. Material is reprocessed if it fails Test B or passes Test C. a) What percentage of the feed material is accepted? b) What percentage of the feed material is reprocessed? c) What percentage of the material which is reprocessed was satisfactory?

Answer 1

a) To find out what percentage of feed material is accepted, firstly we should determine all the possible ways of how material could be accepted

1)Material is satisfactory and it passes both A and B tests

2)Material is non-satisfactor and it passes both A and B tests

Lets calculate pribabilities of each event

$P(1)= 0.85*0.92*0.99=0.77418$

$P(2)=0.15*0.095*0.16=0.00228$

So, the total percentage of the accepted materials is $(P(1)+P(2))*100$% $=77.6$%

b) To find out what percentage of feed material is reprocessed, firstly we should determine all the possible ways of how material could be reprocessed

1) Material is satisfactory and it passes test A and fails test B

2) Material is non-satisfactory and it fails test A and passes test C

Lets calculate pribabilities of each event

$P(1)= 0.85*0.92*0.01=0.0078$

$P(2)=0.15*0.905*0.03=0.0041$

So, the total percentage of the reprocessed materials is $(P(1)+P(2))*100$% $=1.19$%

c) To find out what percentage of reprocessed materials was satisfactory we should use Bayessian probability formula

$P(satisfactory/reprocessed) ={\frac {P(reprocessed/satisfactory)*P(satisfactory)} {P(reprocessed)}} ={\frac {0.0078} {0.0119}}=0.655$

So, 66.5% percentage of reprocessed materials was satisfactory