Question

In: Statistics and Probability

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

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.

Solutions

Expert Solution

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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