Good afternoon. I would like to ask if I have understood and solved the problem correctly. Task:
In one city, 85% of the taxis are green and 15% are blue. A witness to the accident testified that the driver of the blue taxi was the one who caused the accident and then drove away. Tests carried out (under similar lighting conditions) showed that the witness correctly identified the color of the taxi 80% of the time and was wrong 20% of the time. What is the probability that the person responsible for the accident was actually driving a blue taxi?
My attempts at a solution: P(blue taxi)=0.85. P(green taxi)=0.15. P(colour correctly defined)=0.8. P(color incorrectly defined)=0.2. According to the problem, I need to find the conditional probability that the accident was done by a blue taxi, where the condition is that the colour of the taxi is identified correctly.
P(correct color| blue taxi)-?
If say we have 100 taxis in the city. Then 85 taxis will be green and 15 blue.
A header | green | blue | summation |
---|---|---|---|
correct color | 68 | 12 | 80 |
wrong color | 17 | 3 | 20 |
$P(correct |blue)= \frac{P(blue│correct)\cdot P(correct)}{P(blue│correct^c )\cdot P(correct^c )+P(blue│correct)\cdot P(correct)}\\ =\frac{12/80\cdot 20/100}{3/20\cdot 20/100+12/80\cdot 80/100} =\frac{12/400}{3/100+12/100}=\frac{12}{15\cdot 4}=0.2$
am I right?