Question

Of all the cabs, 85 percent are blue and the rest are green. A cab identified in a hit-and-run accident is identified as green. People can correctly identify the color of a cab 80 percent of the time. When surveyed, most Stanford students thought there was an 80 percent chance the cab is actually green. Do you agree? Explain.

Answer 1

No the observation made by most of the Stanford students is incorrect. It is given that :

P(cab is blue) = 0.85

So P(cab is green) = 1- 0.85 = 0.15

Note that : P(A | B) = Probability of an event A given event B

P(people correctly identify the colour of a cab) = P(cab identified as green | it is actually green) = 0.8

We need to find P(it is actually green | cab identified as green)

Now by Bayes Theorem we have,

P(it is actually green | cab identified as green) =  (P(cab identified as green | it is actually green) x P(cab is green)) / (P(cab identified as green))

Now by law of total probabilities,

P(cab identified as green) = P(cab identified as green | it is actually green) x P(cab is green) + P(cab identified as green | it is actually blue) x P(cab is blue) = 0.8 x 0.15 + 0.2 x 0.85 = 0.29

So,

P(it is actually green | cab identified as green) = (0.8 x 0.15) / 0.29 = 0.4137931034

So clearly what most Stanford students thought comes out to be wrong as if just does not depend on the probability of correctly identifying a colour but also the probability of a cab being green.

Thank you. Please rate positively if satisfied!