Respuesta :
Answer:
a) It is not possible
b) Yes, it is possible. It is more likely that the taxi was green. The probability that the taxi was blue given that you saw it was blue is just 0.25
Step-by-step explanation:
Lets call B the event 'the taxi is Blue'. G the event the taxi is 'Green'. AB the event 'The taxi appears to be Blue', and AG the event 'The taxi appears to be Green'. Note that G is the complementary event of B.
a) You lack information to make a desicion. It depends on the proportion of the taxis (in other worlds, the probability for a random taxi to be blue, or P(B) ). What you have is the following
- P(AB | B) = 0.75 (The probability of a blue car being identified as blue)
- P(AB | G) = 0.25 (The probability of a green car being identified as blue)
What we want is P(B | AB) and P(G |AB) = 1 - P(B |AB). That can be calculated using Bayes Theorem, once we know the probabilities of B and G. Without that information it is impossible to know which car is the most likely. You may intuit that blue is the most likely collor for the taxi (because you witness it being blue), but it may not be the case if the proportions of green cars is much higher.
b) As we cocluded in a. We can use Bayes Theorem if we know P(B) and P(G). Since 9 out of 10 cars are green, then P(G) = 0.9 and P(B) = 0.1. Therefore
[tex]P(B|AB) = \frac{P(AB |B)*P(B)}{P(AB)}[/tex]
Using the Theorem of total Probability, we have
[tex]\frac{P(AB |B)*P(B)}{P(AB)} = \frac{P(AB|B)*P(B)}{P(AB|B)*P(B)+P(AB|G)*P(G)} = \frac{0.75*0.1}{0.75*0.1+0.25*0.9} = \frac{0.075}{0.3} = 0.25[/tex]
Thus, there is a probability of 0.25 that the taxi you saw was Blue. It was more likely that it was green.
