What are the chance that person tested positive is really sick? Tax problem The

Question

What are the chance that person tested positive is really sick? Tax problem The blue taxi. In a city with one hundred taxis, 3 are blue and 97 are green. A witness observes a hit-and-run by a taxi at night and recalls that the taxi was blue, so the police arrest the blue taxi driver who was on duty that night. The driver proclaims his innocence and hires you to defend him in court. You hire a scientist to test the witness' ability to distinguish blue and green taxis under conditions similar to the night of accident. The data suggests that the witness sees blue cars as blue 99% of the time and green cars as blue 3% of the time. Write a speech for the jury to give them reasonable doubt about your client's guilt. Your speech need not be longer than the statement of this question. Keep in mind that most jurors have not taken this course, so an illustrative table may be easier for them to understand than fancy formulas.

Explanation / Answer

Here we are given that P( blue ) = 3/100 = 0.03 because there are a total of 3 blue taxis out of 100 total taxis.

Now according to the scientist, the witness sees blue car when it is blue with a probability of 0.99 that is

P( Seen Blue | Blue ) = 0.99

And the witness sees green car as blue with a probability of 0.03.

P( Seen Blue | Green ) = 0.03

Therefore the total probability of seeing blue is computed as:

P( Seen Blue ) = P( Seen Blue | Blue ) *P(Blue ) + P( Seen Blue | Green ) *P(Green)

P( Seen Blue ) = 0.99*0.03 + 0.03*(1-0.03)

P( Seen Blue ) = 0.0588

Now by Bayes conditional probability formula we get:

P( Blue | Seen Blue ) P( Seen Blue ) = P( Seen Blue | Blue ) *P(Blue )

Putting all the values we get:

P( Blue | Seen Blue ) * 0.0588 = 0.99*0.03

Therefore, P( Blue | Seen Blue ) = 0.99*0.03 / 0.0588 = 0.5051

Therefore there is a 0.5051 probability that the car was really blue given that the witness saw it as blue.

As it is approximately equal to 0.5, therefore when the witness sees blue there is an equal chance of the car really being blue or green and therefore one should not be certain that our client was the culprit here.

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