Question 508348: Three machines X, Y, Z are produced respectively 50%, 30% and 20% of the total number of items in a factory. The output of these machines is 3%, 4% and 5% respectively. If an item is selected at random, find
(i) The probability that the item is defective
(ii) Probability that the item is produced by machine X, machine Y, machine Z?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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%
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