# Math Help - Probability Problems...

1. ## Probability Problems...

Suppose we are interested in the condition of a machine that produces a particular item. Let A designate the event "the machine is in good working condition". Let B be the event "the machine is not in good owrking condition". We know from experience that P(A)= 0.90. Further, let C be the event "a defective item is produced". We also know from experience that a defective item is produced from a machine in good operating condition only 1% of the time, while 10% of the time the item is defective if the machine is not in good operationg condition.

-What is the value of P(C|A)?

-What is the value of P(C|B)?

-What is the value of P(B)?

-What is the P(C)?

-Suppose you are givien that an item is defective. What is the probability the defective item was produced by a machine in good operating condition?

2. I will work out P(C|A). As usual, one of the ways of solving these problems, is understanding what is actually being asked, understanding the sample space and relating that so some property or equation that we know.

For P(C|A), we first look at the given part. We are told that the the machine is in good working condition 0.9 percent of the time. This means that 0.9 of the entire sample space the machine is in good working condition (probabilities are no different than raw discrete numbers, such as 9 out of 10 balls are red, and 1 out of 10 balls are blue). So right there we know we are working from a space of 0.9 probability:

$P(C|A)=\frac{X}{0.9}$

With that established, we need to find out, given that we are within that A sample space (which is 0.9 of the total space), what is the probability that it shoots out a defective item. Ok, well we look at the A subspace and we are told that 0.01 of all items within that subspace are defective. Now if we were trying to figure out a probability, what would we do: We would take the number of items that meet some qualification (in this case the number of defective items), and divide that by the number of items in that sample space (in this case the 0.9 - which is space A). Well that's easy:

$P(C|A)=\frac{0.01*0.9}{0.9}$

Where 0.01*0.09 is 0.01 percent OF A (since that is the number of 0.9 that is defective right? Therefore that probability that an item is defective given that the machine is in working condition is 0.01!

Your second question follows the same logic (it is NOT however 1-P(C|A)), your third question is a simple application of the laws of probabilities, and your final question is an application of Baye's theorem that looks (and sounds) creepy. . .but it isn't.

3. I'm not sure I understand the solution to this problem

From the problem we know that

We also know from experience that a defective item is produced from a machine in good operating condition only 1%
Isn't this the same as saying that given the machine is working, 1 out 100 item will be defective? Then P(C|A)=0,01. This is the same result as you got by applying the product rule (right?), but as far as I can see the answer was given in the question. Please help me understand .