Probability random person has disease, if pos. test result

[Angela123]

A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with the probability 0.9. Also, if the person does not have the disease, the test will report that he or she doesn't have it with probability 0.9. Only 10% of the population has the disease in question. If the person is chosen at random from the population and the diagnostic test reports him to have the disease, what is the conditional probability that he does, in fact have the disease?

I did try building a tree for this problem, but my answer didn't match. The correct answer is: 0.5000

 

[Deleted member 4993]

Re: Probability

A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with the probability 0.9. Also, if the person does not have the disease, the test will report that he or she doesn't have it with probability 0.9. Only 10% of the population has the disease in question. If the person is chosen at random from the population and the diagnostic test reports him to have the disease, what is the conditional probability that he does, in fact have the disease?

I did try building a tree for this problem, but my answer didn't match. The correct answer is: 0.5000

Please share your work with us, even if you know it is wrong - so that we know where to begin to help you.

 

[Angela123]

I filled in the blanks.
It would be
90 does not have the disease
90 of those test positive
10 of those test negative
10 have the disease
10 of those test positive
90 of those test negative

So_____out of 100 will test positive, of those_____actually have the disease, so the conditional probability is ..
But it's the first blank I can't fill in...I keep thinking it's 100 out of 100 test positive because I add the 90+10 but I know this is wrong.

 

[galactus]

These things are easy if you draw a diagram. I do not prefer trees myself.

I will go ahead and show you what I mean and then you can use it on other problems. Okey-doke?.

Fill in a chart.

If we use some nice number like 100 or 1000 to represent the total population ,it makes it easier. Since we assumed 100 and 10% have the disease, then we know we can put 10 at the bottom of the infected column. Thus, 90 are not infected. Finish filling it in and we have:

                              infected                              not infected
--------------------------------------------------------------------------------------------
POS                             9                                              9                              18

---------------------------------------------------------------------------------------------

NEG                           1                                               81                             82

-----------------------------------------------------------------------------------------------

                                10                                                90                            100

Now, what is the probability a person is infected given the test shows positive?. Go down the infected column to the POS row and then go across.

We happily get 9/18=.5

You can answer any scenario they throw at you now. What is the prob. the person is infected given they test negative?.

Go down the infected column to the NEG row and go across. We get 1/82. See?.

Makes it easier than trying to lumber through Bayes theorem formula.