Question

A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that was produced by machine B?

Answer 1

Let P( A ), P( B ) and P( D ) be the probabilities that are defined below,

The probability that the items produced by machine A is P( A ).

The probability that the items produced by machine B is P( B ).

The probability that the items is defective is P( D ).

The probability that the item was produced by machine B, if it is found to be defective that is P( B D ),

P( B D )= P( B )P( D B ) P( B )P( D B )+P( A )P( D A ) (1)

The probability that the items produced by machine A is,

P( A )=60% =0.6

The probability that the item is defective, if produced by machine A,

P( D A )=2% =0.02

The probability that the items produced by machine B is,

P( B )=40% =0.4

The probability that the item is defective, if produced by machine B,

P( D B )=1% =0.01

Put these values in equation (1),

P( B D )= ( 0.4×0.01 ) ( 0.4×0.01 )+( 0.6×0.02 ) = 0.4 1.6 = 1 4

Thus, the required probability is 1 4 .