A machine is used in a production process. From past data, it is known that 97% of the time the machine is set up correctly. Furthermore, it is known that if the machine is set up correctly, it produces 95% acceptable (non-defective) items. However, when it is setup incorrectly, it produces only 40% acceptable items. This is Bayes’ Theorem. P(N|C) = .95

The probability that the machine is set up correctly and, The  selected product is non-defective is = 93.35%.

The probability the selected item is non-defective,  the probability that the machine is set up correctly = 98.71%.

The  selected product is non-defective will be; 0.97 x 0.95 = 0.9215.

The probability that the machine is not set up right  ,

The selected product is non-defective is =  (1 - 0.97) x 0.40 = 0.03 x 0.40 = 0.012.

The probability that the selected product is non-defective is the sum of these probabilities:

P = 0.9215 + 0.012 = 0.9335 = 93.35%.

The probability that the selected product is non-defective is 0.9335.

Since, The selected product is definitely non-defective,

The probability that the selected product is non-defective  is 1. This means that the sum of the probability that the machine is set up right and that the selected product is non-defective plus .

The probability that the machine is not set up right ,

The selected product is non-defective is 1.  

0.9335 x a = 1

a = 1 ÷ 0.9335 = 1.0712

Thus,  The probability the selected item is non-defective,  the probability that the machine is set up correctly ,

P = (0.97 x 0.95) x a = (0.97 x 0.95) x 1.0712 = 0.98711 = 98.71%

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The question is not complete and the complete question is;

A machine is used in a production process. From past data, it is known that 97% of

the time the machine is set up correctly. Furthermore, it is known that if the machine

is set up correctly, it produces 95% acceptable (non-defective) items. However, when

it is set up incorrectly, it produces only 40% acceptable items.

a. An item from the production line is selected. What is the probability that the selected item is non-defective?

b. Given that the selected item is non-defective, what is the probability that the machine is set up correctly?

A) 93.35%

B) 98.71%

Step-by-step explanation:

A) The probability that the machine is set up correctly and that the

selected product is non-defective will be; 0.97 x 0.95 = 0.9215

The probability that the machine is not set up right and that the selected product is non-defective is (1-0.97) x 0.40 = 0.03 x 0.40 = 0.012

Thus, the probability that the selected product is non-defective is the sum of these probabilities:

P = 0.9215 + 0.012 = 0.9335 = 93.35%

b. Now, since we know that the selected product is non-defective, then we can find the probability that the machine is set up correctly.

We have seen that the probability that the selected product is non-defective is 0.9335

Hence,

Since the selected product is definitely non-defective, we also

know that the probability that the selected product is non-defective

is 1. This means that the sum of the probability that the machine is set up right and that the selected product is non-defective plus the

probability that the machine is not set up right and that the

selected product is non-defective is 1. This means that;

0.9335 x a = 1

a = 1/0.9335 = 1.0712

Thus, the probability that the machine is set up correctly and the

selected product is non-defective is

calculated as;

P = (0.97 x 0.95) x a = (0.97 x 0.95) x 1.0712 = 0.98711 = 98.71%