When a machine is adjusted properly, 50% of the items it produces are good and 50% are bad. However, the machine is improperly adjusted 10% of the time; in this case, 25% of the items it makes are good and 75% are bad. Suppose that 5 items produced by the machine are selected at random and inspected. If 4 of these items are good (and 1 is bad), what's the probability that the machine was adjusted properly at the time?

Step-by-step explanation:

Probability of good given properly adjusted  P(G/P) = .5

Probability of bad given properly adjusted  P(B/P) = .5

Probability of inappropriately adjusted P(I ) = .1

Probability of properly adjusted P(P) = .4

Probability of good given inappropriately adjusted P( G/I ) = .25

Probability of bad given inappropriately adjusted  P(B/I ) = .75

P( G ) = P(G/P) x  P(P) + P( G/I ) x P(I )

P(P/G) = P(G/P) x  P(P)    /     P(G/P) x  P(P) + P( G/I ) x P(I )

=   .5 x .4  / .5 x .4 + .25 x .1

= .20 / .20 + .025

.20 / .225

20 / 22.5

= 4 / 4.5 .

= 8 / 9 .

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