An engineer is checking on the electric circuit boards and classified them as ex

Question

An engineer is checking on the electric circuit boards and classified them as excellent, acceptable or unacceptable boards. 35% of the boards are excellent and 55% arc acceptable. It was reported that 8% and 25% from the excellent and acceptable boards, respectively failed during the testing. None of the unacceptable boards were tested. An electric circuit board was selected at random. Calculate the probability that the board was excellent and failed. Calculate the probability that the board was failed. Given that the board was failed, calculate the probability that it was classified as excellent.

Explanation / Answer

Given input data

Probability of excellent = 35% i.e. 0.35

Probability of acceptable = 55% i.e. 0.55

Probability of excellent and failed = 8% i.e. 0.08

Probability of acceptable and failed = 25% i.e. 0.25

Probabilty of unacceptable = 100 - (35 + 55) = 10% i.e. = 0.01

a ) From product Rule :

P(Excellent and fails)=P(excellent) * P(fails given excellent)

P(Excellent and fails)=0.35 0.08 = 2.8 %

Probability that board was excellent and failed is 2.8 %

b ) P(fails) = P(excellent) * P(fails given excellent) + P(acceptable) * P(fails given acceptable) +P(unacceptable)*P(fails given unacceptable)

P(fails) = ( 0.35 0.08 ) + ( 0.55 0.25 ) + ( 0.01 1 )

   P(fails) = ( 0.028 ) + ( 0.1375 ) + ( 0.01 )

P(fails) = 0.1755

Probability that board was failed is 17.55 %

c)

from problem b) P(fails ) = 17.55 % i.e. 0.1755

From Bayes Theorm :

P( excellent given fail ) = P (excellent) * P( fails given excellent ) / P(fails)

P( excellent given fail ) = ( 0.35 ) * ( 0.08 ) / ( 0.1755 )

P( excellent given fail ) = ( 0.028 ) / ( 0.1755 )

P( excellent given fail ) = 0.1595

i.e., Given board was failed, probablity that it was classifeid as excellent is 15.95%

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