The following is a homework question that I'm having a lot of trouble with. Despite even possessing the answer to the question, I still can't work it out for myself. If someone could kindly explain and work out the answer for me, I would very much appreciate it. Thank you.

QUESTION:
Approximately 1 in 14 men over the age of 50 has prostate cancer. The level of 'prostate specific antigen' (PSA) is used as a preliminary screening test for prostate cancer.

7% of men with prostate cancer do not have a high level of PSA. These results are known as 'false
negatives'.

75% of those men with a high level of PSA do not have cancer. These results are known as 'false
positives'.

If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?

Hey jacksun.

To get you started I'm going to ask if you can convert the statements into probabilities. For example P(High PSA|No Cancer) and P(Low PSA|Cancer) [Hint: Relate them to false positives and false negatives].

Originally Posted by chiro
Hey jacksun.

To get you started I'm going to ask if you can convert the statements into probabilities. For example P(High PSA|No Cancer) and P(Low PSA|Cancer) [Hint: Relate them to false positives and false negatives].
Is this what you mean:

The probability that a man over the age of 50 has prostate cancer is 1/14

The probability that a man with a high PSA does not have cancer (false negative) is 7/100

The probability that a man with a low PSA does have cancer (false positive) is 75/100 = 3/4

Try writing them in probability form.

It will be easier since you are trying to find P(Has Cancer|Normal PSA) in terms of the actual probability.

Originally Posted by chiro
Try writing them in probability form.

It will be easier since you are trying to find P(Has Cancer|Normal PSA) in terms of the actual probability.
I'm sorry, but I'm afraid I'm just not getting this right now; I'm very confused.

P(Man over age 50 has cancer) = 1/14 = .07

P(No High PSA|Cancer) = .07

P(High PSA|No Cancer) = .75

I think a simple way to handle problems like this is to imagine a specific number of cases (chosen to avoid fractions!). Here, imagine 1400 men over 50. 1400/14= 100 have prostate cancer. 7% of those, 7, will not have a "high level of PSA" which means that 100- 7= 93 men with prostate cancer will have a high level of PSA. Since "75% of those men with a high level of PSA do not have cancer", 25% will have prostate cancer: so 7 is 25%= 1/4 of all men who have a high level of PSA. That, in turn, means that there are a total of 4(7)= 28 men, of our original 1400, have a high level of PSA and, therefore, 1400- 28= 1372 of them have a "normal" level of PSA. And we have already seen that 7 men who have prostate cancer do NOT have a high level of PSA.

Originally Posted by HallsofIvy
so 7 is 25%= 1/4 of all men who have a high level of PSA.
But how can that be - I thought the 7, as stated earlier, represented the men who will NOT have a high level of PSA.

By the way, I know that the final answer to this question is 0.7%

Following on Hallsofivy's suggestion, pick a population size that is easy to work with, 1400. I will be describing the different possibilities by using the letter c and p, upper case C represents having cancer, and upper case P represents having high levels of PSA. For example, Cp, would refer to a person who has cancer but low levels of PSA.

1 in 14 means there are 100 people with cancer - so Cp = 7/100, therefore CP = 93/100.

75% of people with high PSA levels do not have cancer, so 25% are CP, therefore the total high PSA numbers is 93/0.25=372, hence cP=279.

So the probability that Cp occurs is 7/(1400-372), which is approximately 0.0068.

Cheers