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

Let $S$ denote the event "feed material is satisfactory".

Let $A$ denote the event "feed material passes Test A".

Let $B$ denote the event "feed material passes Test B".

Let $C$ denote the event "feed material passes Test C".

Given

$$P(S)=0.85, P(\overline{S})=1-0.85=0.15$$

$$P(A|S)=0.92, P(A|\overline{S})=0.095$$

a) The probability that feed material is satisfactory, and passes both Test A and Test B is

$$P(S\cap A\cap B)=0.85(0.92)(0.99)=0.77418$$

The probability that feed material is not satisfactory, and passes both Test A and Test B is

$$P(\overline{S}\cap A\cap B)=0.15(0.095)(0.16)=0.00228$$

The probability that feed material is accepted

$$P(accepted)=0.77418+0.00228=0.77646$$

$77.646\%$ of the feed material is accepted.

b) The probability that feed material is satisfactory, passes Test A and fails Test B is

$$P(S\cap A\cap \overline{B})=0.85(0.92)(0.01)=0.00782$$

The probability that feed material is not satisfactory, passes Test A and fails Test B is

$$P(\overline{S}\cap A\cap \overline{B})=0.15(0.92)(0.84)=0.11592$$

The probability that feed material is satisfactory, fails Test A and passes Test C is

$$P(S\cap \overline{A}\cap C)=0.85(0.08)(0.82)=0.05576$$

The probability that feed material is not satisfactory, fails Test A and passes Test C is

$$P(\overline{S}\cap \overline{A}\cap C)=0.15(0.905)(0.03)=0.0040725$$

The probability that feed material is reprocessed

$$P(reprocessed)=0.00782+0.11592$$ $$+0.05576+0.0040725=0.1835725$$

$18.35725\%$ of the feed material is reprocessed.

c) The probability that feed material is satisfactory, and is reprocessed

$$P(S\cap reprocessed)=0.00782+0.05576$$ $$=0.06358$$

$$\dfrac{0.06358}{0.1835725}\cdot100\%\approx34.6348\%$$

$34.6348\%$ of the material which is reprocessed was satisfactory.