Question

A factory has four machines that produce the same item. Machines 1 and 2 each produce 20% of the total output, while machines 3 and 4 are larger and each produces 30% of the total output. It is known that 6% of machine 1's output is defective, while machine 2 produces 5% defective items, and machines 3 and 4 each produces 8% defective items. a. An item is chosen at random from the output of the factory. What is the probability that this item is defective? b. Given that the item selected in part a. is defective, what is the probability that it was produced by machine 1? c. Given that the item selected in part a. is defective, what is the probability that it was produced by machine 2? d. Given that the item selected in part a. is defective, what is the probability that it was produced by machine 3? e. Given that the item selected in part a. is defective, what is the probability that it was produced by machine 4? f. What is the total of the probabilities in parts b.-e.?

Answer 1

The answers are a. 0.27 b. 0.2 c. 0.2 d. 0.3 e. 0.3 f. 1

Explanation:

Total output = 100% = 1

Total defective = 6% + 5% + 8% + 8% = 27% = 27

a. Prob of defective item = total defective/total output

= 27/100

= 0.27

b. Prob of defective from machine 1 = 6/27 =0.2222

~ 0.2

c. Prob of item defective from machine2 = 5/27 = 0.1852

~0.2

d. Prob of de defective from machine 3 = 8/27 = 0.2963

~0.3

e. Prob of defective from machine 4

= 8/27 = 0.2963

~ 0.3

f. Sum of prob from b-e

0.2 + 0.2 + 0.3 + 0.3

= 1