Test your mettle — a probability theory problem

Some of the most formidable and extremely interesting problems are in probability theory of exams like IITJEE, ISI entrance, CMI entrance etc. Even otherwise, probability theory quickly  becomes abstract, whether you study as an engineering student or as an applied mathematics student or as a pure mathematics or statistics student.

Let me present to you a seemingly simple question which test your conceptual understanding of probability theory basics.

Question: (Parzen). Suppose there exists a (fictitious) test for cancer with the following properties. Let A be the event that the test states that tested person has cancer and B be the event that the person has cancer.

It is known that P(A|B)=P(A^{'}|B^{'})=0.95 and P(B)=0.005. Is this test a good test?

Solution:

Before reading the solution, just give it a shot! And, then, after that if you see the solution, it will erase your doubts.

Now, A^{'} is the event that the test states person is free from cancer.

Also, B^{'} is the event that person is free from cancer.

To answer the question we should like to know the likelihood that a person actually has cancer if the test so states, that is, 

P(B|A). Hence,

P(B|A) = \frac {P(B)P(A|B)}{P(A|B)P(B)+P(A|B^{'})P(B^{'})}

The above equals \frac {(0.005)(0.95)}{(0.95)(0.095)+(0.05)(0.995)} = 0.087

Hence, in only 8.7 percent of the cases where the tests are positive will the person actually have cancer. This test has a very high false alarm rate and in this sense cannot be regarded as a good one. The fact that , initially, the test seems like a good test is not surprising given that P(A|B)=P(A^{'}|B^{'})=0.95. However, when the scarcity of cancer in the general public is considered, the test is found to be vacuous.

More later…

Please do post your comments, suggestions …

Nalin Pithwa