# conditional probability, provided that the colour is correctly defined.

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.

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?

• No. Your table uses the word "blue" to mean "actually blue". But your calculations confuse this with "says blue" May 29, 2022 at 17:32
• @Henry then how is it right? May 29, 2022 at 17:42
• Looking at your table, there are $17+12=29$ cases in which the taxi would be identified as blue (the given condition is that the taxi is identified as blue, not that the identification is correct.) Out of these $29$ cases, there are $12$ in which the taxi actually is blue.
– WW1
May 29, 2022 at 18:07
• @WW1 so that is the answer? 12/29. Then I get it. May 29, 2022 at 19:33
• Similar but slightly different: math.stackexchange.com/questions/66263/… May 29, 2022 at 22:31

... 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.

$$\def\={\,{=}\,}$$No, you need the conditional probability that the accident was done by an actual blue taxi, where the condition is that the taxi was identified as blue.

The event of "correct" is the event of "it is actually the colour that it was identified as being". This is a union of "Actually blue and identified as blue, or actually green and identified as green".

Since we are told that the probability for being correct is independent of the actual color, then we have been provided that: $$\mathsf P(I\=b\mid A\=b)=0.80\tag{~=\mathsf P(I\=g\mid A\=g)~}$$

Likewise, $$\mathsf P(I\=b\mid A\=g)=0.20\tag{~=\mathsf P(I\=g\mid A\=b)~}$$

And of course: $$\mathsf P(A\=g)=0.85, \mathsf P(A\=b)=0.15$$

So you may now use Bayes' Rule to find the probability that the taxi is actually blue given that it was identified as blue: $$\mathsf P(A\=b\mid I\=b)$$

• Remark: In questions of this type, it is always advisable not to think in terms of being "correct", but rather in terms of the "identified value" as compared to the "identified value". May 31, 2022 at 1:04