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Math: Probability Question

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Kilgore Trout

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Little math question from the internet:

Tango City has exactly 100 taxicabs.
80 taxis are blue and 20 taxis are green.
Every single taxicab in Tango City is equally likely to be involved in an accident.

Last night a taxi caused an accident and took off.

Question 1:
If an 80 % reliable witness says that a green taxi caused the accident what is the probability that he is right?

Question 2:
If the same 80 % reliable witness said a blue taxi was involved in the accident what is the probability that he is right?
 
W

wawa

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Jan 15, 2004
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I believe #1 is 16% and #2 is 64%. And I am probably wrong as this seems too easy. LOL!!
 
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Kilgore Trout

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wawa said:
I believe #1 is 16% and #2 is 64%. And I am probably wrong as this seems too easy. LOL!!
Click to expand...
According to my math, that is the wrong answer.
I modified the original question very slightly; but, anyway I'll post solution later.
 
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danmand

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Kilgore Trout

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gww

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Somewhere but not here.
Given the same witness can't make up his mind which colour it was what are the odds of it actually being a taxi involved in the accident or what are the odds there was an actual accident? Maybe it was more than one taxi?
 
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Kilgore Trout

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Maybe this question is too tough for Terb.
I'm surprised Danmand didn't get it. He solved my last little vexing math problem in 2 seconds flat.
Probability questions can be notoriously tricky to solve though and need some interpreting.

Someone will eventually come along with the right answer though and after reading his solution the next poster will say that was an easy question.



I'll give you a little hint if you want it.

If the witness was 90% reliable instead of only 80% reliable:

If he said a green taxi caused the accident there is a 69.2 % chance that he is right
If he said a blue taxi caused the accident there is a 97.3 % chance that he is right
 
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Kilgore Trout

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Okay, the solution is posted here and the challenge is to incorporate base rates into the working out of probabilities.
A lot of people assume base rates are irrelevant; but, in probability they are important.

https://plus.maths.org/content/solution-taxi-problem


http://documents.routledge-interact...848724167/Chapter 13 CASE STUDY Base-rate.pdf

I modified the taxi numbers slightly from 85 % : 15 % to 80% : 20%.
So, plugging in the new numbers into the given probability table, if the witness thought the taxi was blue there is a 94.1% chance he is right and if he said the car was green there's a 50% chance he's right.
 
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EJunkie

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Your argument doesn't work when you ascribed an 80% reliability factory for the witness (something not in the two scenarios you linked). That overrides the rest because the witness is 80% likely to be correct.
 
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frankcastle

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Don't you multiply probability the person is correct 80% by probability that it is the car of specific colour

So 80% x 20% = 16% is #1

80% x 80% = 64% is #2
 
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Kilgore Trout

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EJunkie said:
Your argument doesn't work when you ascribed an 80% reliability factory for the witness (something not in the two scenarios you linked). That overrides the rest because the witness is 80% likely to be correct.
Click to expand...
It's not my argument. It's the solution put forward by experts in their probability contingency tables


===========================Probability Contingency Table============================


= =========== ======= Blue Taxi Involved ========== Green Taxi Involved

Number of cases ============ 80 ===================== 20============================


Witness Report

Blue Taxi============== .8 * 80 = 64 =============== .2 * 20 = 4 ============= 68 Blue Cases from witness report

Green Taxi============= .2 * 80 = 16 =============== .8 * 20 = 16 ============32 Green Cases from witness report

From the table above you have 100 outcomes 64 + 4+ 16+16= 100

If he reported Blue he would have been right 64 times and wrong 4 times for a total of 94.1 per cent right
If he reported Green he would be right 16 times and wrong 16 times for a total of 50 per cent right.
 
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IM469

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Kilgore Trout said:
It's not my argument. It's the solution put forward by experts in their probability contingency tables
If he reported Blue he would have been right 64 times and wrong 4 times for a total of 94.1 per cent right
If he reported Green he would be right 16 times and wrong 16 times for a total of 50 per cent right.
Click to expand...
I think the 'experts' are chasing their tails done a toilet over thinking this.

If a witness reports a green taxi and 80% of the time it is a green taxi ... thats 4 out of 5 times he is correct..... so what's this crap he's right 50% ? :confused:
 
B

Big Rig

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IM469 said:
I think the 'experts' are chasing their tails done a toilet over thinking this.

If a witness reports a green taxi and 80% of the time it is a green taxi ... thats 4 out of 5 times he is correct..... so what's this crap he's right 50% ? :confused:
Click to expand...
If the witness is 100% reliable are not the odds 100% he is correct?


This is a conditional probability question.

You are ignoring what is called the base rate IE taxi probability

Use Bayes theorem. I think the answer is 94% by my calculations

And with Bayes math if the witness is 100% correct you get infinity for the answer or something like that (I am just a truck driver )
 
R

r__d_ott

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Yes, this is a Bayesian conditional probability. Made famous in Daniel Kahneman's book, I think.
 
K

Kilgore Trout

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Big Rig said:
If the witness is 100% reliable are not the odds 100% he is correct?


This is a conditional probability.

You are ignoring what is called the base rate.

Use Bayes theorem
Click to expand...
Big Rig is 100% correct.
Bayes' Theorem handles probability situations like this where there is accurate knowledge of background environmental circumstances.

And apparently almost all people have base rate blindness when they try to figure out odds in a situation like with the taxis in this case.

Another example.
If we know for a fact that exactly one in a thousand people are heroin addicts, (base rate), and someone comes up with a 99% reliable test for heroin addiction; - that test will yield a false positive result 90% of the time.
If he did the test a thousand times, he would have 10 positive heroin addiction results instead of the expected one.
If the test were 100% accurate, there would be no problem with false positives.

https://en.wikipedia.org/wiki/Bayes'_theorem
 
B

benstt

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This is a more mathematical description of what is going on in here.

https://plus.maths.org/content/solution-taxi-problem-revisited

The key to understanding this comes to the difference in these statements:

The probability that the witness can correctly identify a given green taxi as green is 80%. (That is his reliability, given you know what colour the car actually is.)

The probability that the offending taxi is actually green given the witness identifies it is green. (ie the probability that he is right about the offending taxi colour, given he thinks it was green.)

The latter question needs to account properly for four situations. 1. the the taxi is actually green (and he is correctly identifying is as green) 2. the taxi is actually green (and he mistakenly identifies it as blue) 3. the taxi is actually blue (and he correctly identifies is as blue), and 4. the taxi is actually blue (and he wrongly identifies it as green.)

If he identifies it as green, it could be situation 1 or 4. The underlying prevalance of blue vs green taxis comes into play when doing that accounting. The earlier answer described it in terms of contingency tables, which are the same as the four situations i laid out. I prefer the mathematical description in the answer I posted, which uses Bayes formula.
 
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