Answer:
0.1739 = 17.39% probability that the cab actually was blue
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
![P(B|A) = \frac{P(A \cap B)}{P(A)}](https://tex.z-dn.net/?f=P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D)
In which
P(B|A) is the probability of event B happening, given that A happened.
is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Witness asserts the cab is blue.
Event B: The cab is blue.
Probability of a witness assessing that a cab is blue.
20% of 95%(yellow cab, witness assesses it is blue).
80% of 5%(blue cab, witness assesses it is blue). So
![P(A) = 0.2*0.95 + 0.8*0.05 = 0.23](https://tex.z-dn.net/?f=P%28A%29%20%3D%200.2%2A0.95%20%2B%200.8%2A0.05%20%3D%200.23)
Probability of being blue and the witness assessing that it is blue.
80% of 5%. So
![P(A \cap B) = 0.8*0.05 = 0.04](https://tex.z-dn.net/?f=P%28A%20%5Ccap%20B%29%20%3D%200.8%2A0.05%20%3D%200.04)
What is the probability that the cab actually was blue?
![P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.04}{0.23} = 0.1739](https://tex.z-dn.net/?f=P%28B%7CA%29%20%3D%20%5Cfrac%7BP%28A%20%5Ccap%20B%29%7D%7BP%28A%29%7D%20%3D%20%5Cfrac%7B0.04%7D%7B0.23%7D%20%3D%200.1739)
0.1739 = 17.39% probability that the cab actually was blue