Expert: Ahmed Salami - 9/29/2010

Question
1. Two manufacturing plants produce similar parts. Plant 1 produces 1000 parts, 100 of which are defective. Plant 2 produce 2000 parts, 150 of which are defective. A part is selected at random and found to be defective. What is the probability that it came from plant 1?

2. A certain class of highway bridge has 10 supporting columns, each of which is inspected annually for weakness. Experience shows that, with annual inspection, the probability of a column weakening seriously during the year is 10 power of -2, independent of the condition of the other columns. Determine the probability that two bad columns would occur.  

Answer
Hi Mohd,
1) The easiest way to go about it would be to consider that there are 100 such parts (that are defective from plant 1) in a total of 3000 parts and the probability is 100/3000 = 1/30.
Or you could multiply the probabilities that the chosen part is from plant 1 and then that it is defective which is a probability of
1000/3000 x 100/1000 = 100/3000
= 1/30

2) The probability that a column weakens is 10^-2 = 1/100 = 0.01
The probability that it doesnt weaken is then 1 - 0.01 = 0.99
The probability that two particular columns weaken (and that the remaining eight dont weaken) = (0.01)^2 . (0.99)^8
But the probability we require is that of any two random columns weakening and so we need to know the number of random pairs we can get from 10 columns. In combinatorics, this is given by 10C2 (read '10 combination 2')
10C2 = 10!/2!(10-2)!
= 10!/2!8!
where 10! = 10x9x8x7x6x5x4x3x2x1
Therefore, the probability that any two columns weaken is
(10!/2!8!) . (0.01)^2 . (0.99)^8
= 0.0042
which is less than 5%

Regards