Taken for a ride

[En_Route]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red?

[hismikeness]

Crimson, red, they're all the same in the Crayola 16. Those are the colors by which I live.

Would seem to me if you are correct about Crimson 80% of the time, but there's only 15% actually being Crimson that most likely the car was red. Why, I don't know because half way through typing that I realized I red it wrong the first time.  ???

[Stevil]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red?

This is ambiguous.
When you say "I can be expected to correctly identify a crimson taxi 80% of the time"
What is meant by "of the time"?
Does this mean that off all the cars that you see as being crimson, 80% of them actually are crimson where as 20% of them are actually red?
Is this in the current environment of 85% red cars and 15% crimson cars?
So if 100 cars pass you by, 85 of them will be red, 15 will be crimson, but you will see 18.75 crimson (15 which are crimson, 3.75 which are red - Note that 15 is 80% of 18.75), which means of the 85 red cars you have seen 4.41% of them as crimson.

Which is all just noise.

If we get back to your statement 
"I can be expected to correctly identify a crimson taxi 80% of the time"
Well, if you say that you saw a crimson car, and 80% of the time you are correct, then I would suggest the car is probably crimson (80% likely to be)

[OldGit]

I'm not riding in either of 'em until they've been inspected.

[Ali]

I always thought crimson was red.

[En_Route]

I always thought crimson was red.

Yes, but red is not crimson.

[The Black Jester]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red?

Out of 100 cars, 85 of which are actually red, 15 of which are actually crimson, you would identify more crimson cars than you actually saw: 20% of the 85 red cabs you saw you would mis-identify as crimson, if I'm reading this correctly, which means you would identify 32 cars as crimson, only 15 of which actually were crimson.  17 would be red.  So there is a slight chance in favor of the cab actually being red.  That's my stab in the dark.

[The Magic Pudding]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red?

Out of 100 cars, 85 of which are actually red, 15 of which are actually crimson, you would identify more crimson cars than you actually saw: 20% of the 85 red cabs you saw you would mis-identify as crimson, if I'm reading this correctly, which means you would identify 32 cars as crimson, only 15 of which actually were crimson.  17 would be red.  So there is a slight chance in favor of the cab actually being red.  That's my stab in the dark.

That sounds sensible but it has been my experience with these horrid Enroute things that 100% of the time they aren't sensible at all.

This guy has aberrant vision and self expression, he's an unreliable witness.
A crimson taxi goes past and he's 80% likely to say it's crimson, so 3% of cars will be mistakenly identified as red.

There's probably a trick in the "other 20%
20% of what? 
Does 3% of all cars represent 80% of the errors?
If so the error rate is 3.75% of which only 0.75% applies to reds identified as crimson.
If this is the case it's 94% likely a crimson is crimson.

[Stevil]

This guy has aberrant vision and self expression, he's an unreliable witness.
A crimson taxi goes past and he's 80% likely to say it's crimson, so 3% of cars will be mistakenly identified as red.

But at no point does he state that crimson cars are mistakenly identified as red.

"I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi."

[The Black Jester]

So, En_Route, at this point may we humbly ask for the correct solution?

[The Magic Pudding]

So, En_Route, at this point may we humbly ask for the correct solution?

Why don't we run this guy down, he won't know what hit him.

[Ali]

So, En_Route, at this point may we humbly ask for the correct solution?

Why don't we run this guy down, he won't know what hit him.

We'll use a red car and he'll tell the cops it was crimson.  Yes, it's perfect.

[Stevil]

So, En_Route, at this point may we humbly ask for the correct solution?

Why don't we run this guy down, he won't know what hit him.

We'll use a red car and he'll tell the cops it was crimson.  Yes, it's perfect.

Don't be so sure,
By my calculations, he only mistakes about 4% of the red cars for crimson. So most likely he will think your red car is a red car 96% of the time, unless he has some other issues he hasn't revealed to us.
If you use a crimson car then he is only 80% sure it is crimson though, so there is much more doubt.

So the correct answer is,
use a crimson car to run him over with LOL

[En_Route]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red

Out of 100 cars, 85 of which are actually red, 15 of which are actually crimson, you would identify more crimson cars than you actually saw: 20% of the 85 red cabs you saw you would mis-identify as crimson, if I'm reading this correctly, which means you would identify 32 cars as crimson, only 15 of which actually were crimson.  17 would be red.  So there is a slight chance in favor of the cab actually being red.  That's my stab in the dark.

D'accord.

[Stevil]

Where I live there are two taxi firms, Red Cabs which have 85% of taxis on the road and Crimson Cabs which have the other 15%. Both firms'  taxis sport the colour of their  owners. Last night I saw a cab crash and identified it as a Crimson taxi. However, my vision is not wholly reliable at night; I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi. Was the taxi I saw more likely to have been crimson or red

Out of 100 cars, 85 of which are actually red, 15 of which are actually crimson, you would identify more crimson cars than you actually saw: 20% of the 85 red cabs you saw you would mis-identify as crimson, if I'm reading this correctly, which means you would identify 32 cars as crimson, only 15 of which actually were crimson.  17 would be red.  So there is a slight chance in favor of the cab actually being red.  That's my stab in the dark.

D'accord.

So that is the correct answer?

But I don't understand.

You stated
"I can be expected to correctly identify a crimson taxi 80% of the time; the other 20% of the time I will mistake a red taxi for a crimson taxi."
But you haven't stated that you mistake red taxis 20% of the time for crimson.

The first part of above
"I can be expected to correctly identify a crimson taxi 80% of the time;"
This could be either
1. out of all the crimson taxis that pass you by, you only recognise 80% of them as crimson
or
2. out of all the crimson taxis that you recognise only 80% of them are actually crimson

But when you state 
"the other 20% of the time I will mistake a red taxi for a crimson taxi"
"...the other..." implies option 2 from above.


If The Black Jester's statement was correct then of the 100 taxis 85 which are red and 15 which are crimson, you see 32 crimson cars, 17 which are actually red and 15 which are actually crimson.
However, how does this fit with your statement "I can be expected to correctly identify a crimson taxi 80% of the time;"
In TBJs answer you are only correctly identifying a crimson taxi 46.9% of the time, not 80% of the time.