8. A new AIDS test is available. It is said to be 99% accurate for those with AI

Question

8. A new AIDS test is available. It is said to be 99% accurate for those with AIDS but the false positive rate is 8%. If 0.8% of a particular population has AIDS, what is the probability that a randomly chosen person in that population

(a) Draw the decision tree

(b) has AIDS, given a positive test result?

(c) does not have AIDS, given a positive result?

I know this uses Bayes' Theorem. But I'm not quite sure how to set it up. Please explain how to set it up and how to tackle (solve) other Bayes' Theorem problems for future practice.

Explanation / Answer

(b)

Let D denote that the person has AIDS
Dc denotes the event that the person doesn't have AIDS
Let Y be the event that the test gives the positive result (person has AIDS)

Now,
P(D)=0.008
P(Y|D)=0.99
P(Y|Dc)=0.08
We have to find P(D|Y).

Using Bayes' theorem
P(D|Y)=P(Y|D)P(D)/P(Y)
P(Y)=P(Y(DDc))=P(YD)+P(YDc)=P(Y|D)P(D)+P(Y|Dc)P(Dc)
given values, we have
P(Y)=0.99*0.008+0.08*0.992=0.08728
Therefore,
P(D|Y)=0.99*0.008/0.08728
=0.00792/0.08728
=0.0907

(c)1-p(ans in (b))

= 1-0.0907

=0.9093

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