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?
Answer:
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%
