Question

Consider a medical diagnosis problem in which there are two alternative hypotheses: 1.
that the patient has a particular form of cancer (+) and 2. That the patient does not (-). A
patient takes a lab test and the result comes back positive. The test returns a correct
positive result in only 98% of the cases in which the disease is actually present, and a
correct negative result in only 97% of the cases in which we disease is not present.
Furthermore, .008 of the entire population have this cancer. Determine whether the
patient has Cancer or not using
MAP hypothesis.​

Answer 1

Answer:

qwertyuioplkajh64949*

Answer 2

Answer:

P(cancer)    =  .008

 P(!cancer)   =  .992 =  1 - P(cancer)

 P(+|cancer)  =  .98

 P(-|cancer)  =  .02  =  1 - P(+|cancer)

 P(-|!cancer) =  .97

 P(+|!cancer) =  .03  =  1 - P(-|~cancer)

 (Recall that the exclamation sign "!" means "not")

Explanation:

We want to calculate the probability of cancer given that the test

result is positive.  Or in probability notation, we want P(cancer|+).

However, we are not given P(cancer|+) directly.  This is only an

inconvenience, though, since we can calculate P(cancer|+) using Bayes

rule.  Bayes rule states that:

 P(cancer|+) = P(+|cancer) x P(cancer) / P(+).

We are given both P(+|cancer) and P(cancer) so calculating the

numerator is straight forward.  The denominator we don't have

directly, but we can calculate it pretty easily using the law of total

probability, or "marginalization" as it was called in lecture.  The

law of total probability just says that the probability of testing

positive is:

 P(+) = P(+|cancer) x P(cancer) + P(+|~cancer) x P(~cancer).

So now just plug and chug since we have all the required

numbers:

 P(cancer|+)

    = P(+|cancer) x P(cancer) /

      ( P(+|cancer) x P(cancer) + P(+|!cancer) x P(!cancer) ).

    = .98 x .008 / ( .98 x .008 + .03 x .992 )

    = .0078 / ( .0078 + .0298 )

    = .21

So even though the patient tested positve, the probability that he has

cancer is only 21%.  How can we make sense of this, especially since

the cancer test is quite accurate?  The answer lies in the fact that

the prior probability of having the cancer is extremely low (0.008).

What Bayes rule is telling us here is that because the prior is so

low, we need a lot of evidence to convince us that the patient really

has cancer.  One test result is simply not enough.