Question

Three machines A, B and C produce respectively 50%, 30% and 20% of the total number of items of a factory. The percentage of defective output of these machines are 3%, 4% and 5% respectively. if an item selected at random, find the probability that the item is non-defective.

Answer 1

3 machines are x, y, and z.
x produces 50% of the output.
y produces 30% of the output.
z produces 20%of the output.
output of machine x is 3% defective.
output of machine y is 4% defective
output of machine z is 5% defective.
1. find the probability that an item is defective.
2. find the probability that the defective part is from machine x, machine y, machine z.
defective parts from machine x are .5 * .03 = .015 of the total.
defective parts from machine y are .3 * .04 = .012 of the total.
defective parts from machine z are .2 * .05 = .01 of the total.
total defective parts from all 3 machines = .015 + .012 + .01 = .037 of the total.
if a part is found to be defective, then:
probability that it comes from machine x = .015 / .037 = .4054054054
probability that it comes from machine y = .012 / .037 = .3243243243
probability that it comes from machine z = .01 / .037 = .2702702703
sum of the probabilities from machine x and y and z = 1 as it should be.
given that the part is defective, the probabilities are as stated above rounded to 3 decimal places

406 for machine x
.324 for machine y
.270 for machine z
as a test, assume that the output from all 3 machines was 3000 parts.
machine x produces 1500 of them.
machine y produces 900 of them.
machine z produces 600 of them.

defective parts from machine x = .03 * 1500 = 45
defective parts from machine y = .04 * 900 = 36
defective parts from machine z = .05 * 600 = 30
total defective parts = 45 + 36 + 30 = 111
total parts = 3000
defective / total = 111/3000 = .037
machine x defective divided by total defective = 45/111 = .4054054054
machine y defective divided by total defective = 36/111 = .3243243243
machine z defective divided by total defective = 30/111 = .2702702703
numbers and rates check out.
percent is equal to rate * 100%.
example:
.270 * 100% = 27%