Here is the crux of the problem:
A man gets hit-and-run by a taxi. An eye-witness says the taxi is orange. 85% of the taxis in town are crimson and 15% are orange. The accident occurs at night and the probability that the eye-witness mistook either for the other is 0.2. The man sues the ORANGE company. But, the judge will decide in favor of the man only if the evidence points to a probability of more than 0.7. How does the judge find? Why?
I'm not sure whether this problem requires conditional probability. The way I see it is this:
The probability that witness is confused = 0.2
The probability that the witness is wrong = 0.2 x 0.85 = 0.17
Since the Judge requires 70% probability, she will find in favor of the man, irrespective of what the proportion of taxis is in the city. I say this as the maximum probability that the eye-witness is WRONG is 0.2. Hence, the judge can be AT LEAST 80% sure about the orange company being guilty.
Does my argument sound logical? Is there something I'm ignoring? If I'm right, that means there's no point in including details about the proportion of taxis. Thats whats bugging me. Would Bayes' theorem apply to this? In that case, am I wrong? PLEASE REPLY!