Test background Suppose there is a new test to determine if someone has what it

Question

Test background Suppose there is a new test to determine if someone has what it takes to be a comic book artist. We know that only 1/50 people have what it takes to be a comic book artist. When someone has what it takes, the test is positive 98% of the time. When someone doesn't have what it takes, the test is negative 96% of the time.

Part A

Suppose you are tested, and the results say you DO have what it takes to be a comic book artist, what are the chances that you really do have what it takes? Round your answer to four decimal places.

Part B

Suppose you are tested, and the results say you DON'T have what it takes to be a comic book artist, what are the chances that you really do have what it takes? Please give your answer to four decimal points.

Explanation / Answer

We are given here that:

P( artist ) = 1/50 = 0.02, therefore P( not artist ) = 0.98

Also, we are given that:
P( positive | artist ) = 0.98, therefore P( negative | artist ) = 0.02
P( negative | not artist ) = 0.96, therefore P( positive | not artist ) = 0.04

Part A:

Using law of total addition we get here:
P( positive ) = P( positive | artist )P( artist ) + P( positive | not artist )P( not artist ) = 0.98*0.02 + 0.04*0.98 = 0.0588

Therefore, P( artist | positive ) = P( positive | artist )P( artist ) / P( positive ) = 0.98*0.02 / 0.0588 = 1/3

Therefore 1/3 = 0.3333 is the required probability here.

Part B:

We computed above that P( positive ) =  0.0588, therefore P( negative ) = 1 - 0.0588 = 0.9412

Therefore, using Bayes theorem, we get here:

P( artist | negative ) = P( negative | artist ) P( artist ) / P( negative )

P( not artist | negative ) = 0.02*0.98 / 0.9412 = 0.0208

Therefore 0.0208 is the required probability here.

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