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Topic: Black and White Taxis (Read 658 times) |
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ThudnBlunder
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Black and White Taxis
« on: Nov 4th, 2003, 11:19am » |
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A witness observes an accident involving a taxi in London. The witness asserts that the taxi is black. From previous research it is known that witnesses are correct only 80% of the time when making such statements. The police also know that 85% of the taxis in London are black, the other 15% being white.
1) What is the probability that a black taxi was involved in the crime?
2) If the witness reported a white taxi being involved, what is the probability that he/she is correct?
(Not sure if this belongs in Medium.)
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| « Last Edit: Nov 5th, 2003, 6:29am by ThudnBlunder » |
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aero_guy
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Re: Black and White Taxis
« Reply #1 on: Nov 4th, 2003, 11:23am » |
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This sounds a whole lot like one of the Willy STD problems
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ThudnBlunder
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Re: Black and White Taxis
« Reply #2 on: Nov 4th, 2003, 11:39am » |
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A willy's STD problems?? 
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towr
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Re: Black and White Taxis
« Reply #3 on: Nov 4th, 2003, 12:04pm » |
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It's a more than one-year old thread, but it's on the main site as well.. here's the thread:WILLYWUTANG HAS STDS?
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| « Last Edit: Nov 4th, 2003, 12:05pm by towr » |
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Nick McGrath
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Of course, it is well known that London taxis are multicolored and many of them are mobile billboards.
However, given that the information provided is correct, we have
[(1) p(black)=p(taxi black/witness correct)+(p(taxi white/witness incorrect)
=>p(black)=.85*.8+.15*.2=.71
(2) p(white)=p(white/correct)+p(black/incorrect)
=>p(white)=.15*.8+.85*.2= .29
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ThudnBlunder
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Re: Black and White Taxis
« Reply #5 on: Nov 4th, 2003, 9:14pm » |
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Quote:Of course, it is well known that London taxis are multicolored and many of them are mobile billboards.
However, given that the information provided is correct, we have
[(1) p(black)=p(taxi black/witness correct)+(p(taxi white/witness incorrect)
=>p(black)=.85*.8+.15*.2=.71
(2) p(white)=p(white/correct)+p(black/incorrect)
=>p(white)=.15*.8+.85*.2= .29
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You were doing well until the 3rd line. 
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| « Last Edit: Nov 4th, 2003, 11:21pm by ThudnBlunder » |
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NickMcG
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Re: Black and White Taxis
« Reply #6 on: Nov 5th, 2003, 6:31pm » |
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Aaah. Thought it looked too easy.
Another try.
{1) Witness testifies taxi is black, we have 2 possibilities
a) Taxi black;correctly identified p(b,c)= .85*.8 = .68
b) Taxi white; wrongly identified p(w,w)= .15*.2 = .03
Then by Bayes' p(black) = .68/(.68+.03)= 68/71=~0.958
2) Here I get confused:
Witness testifies taxi is white
a) Taxi white; witness correct p(w,c) = .15*.8 = .12
b) Taxi black, wrongly identified p(b,w) = .85* .2 = .17
And p(white) = .12/(.12+.17) = 12/29 =~ 0.414
from which we conclude probability witness is correct is 0.414
However, we are also told p(witness is correct) = 0.8
Help!}
BTW, as a newcomer to this site I have not figured out how to hide text. How do you do that?
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Icarus
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Re: Black and White Taxis
« Reply #7 on: Nov 5th, 2003, 6:42pm » |
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Welcome to the forum! You use the hide tags: [hide] [/hide]. I would not bother hiding on this thread at this point however. Anyone read this far down past towr's link will be looking at everything anyway, so you might as well make it easy for them.
As for your logical dilemma, I'll leave you to T&B's tender mercies. 
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ThudnBlunder
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NickMcG, the correct identifications of white taxis are more than outweighed by the false identifications of black taxis. (Hence the witness's evidence is of no practical value.)
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| « Last Edit: Nov 5th, 2003, 11:27pm by ThudnBlunder » |
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NickMcG
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Re: Black and White Taxis
« Reply #9 on: Nov 6th, 2003, 6:12am » |
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Surely it is wrong to say that the witness has no practical value.
With him, the police know with 41 per cent probably that the offending taxi was white. Without him, there is only a 15 per cent chance.
If I were in charge of the inquiry it would certainly make me divert a couple of detectives to look at white cabs.
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ThudnBlunder
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Re: Black and White Taxis
« Reply #10 on: Nov 6th, 2003, 6:33am » |
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Quote:| If I were in charge of the inquiry it would certainly make me divert a couple of detectives to look at white cabs. |
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In terms of allocating resources, I agree that it is somewhat useful.
In terms of deciding whether it was a black taxi or a white taxi, one may as well toss a (Belgian Euro) coin. 
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| « Last Edit: Nov 6th, 2003, 6:57am by ThudnBlunder » |
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rmsgrey
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Re: Black and White Taxis
« Reply #11 on: Nov 14th, 2003, 7:45am » |
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The witness stating the taxi is white is only useless if it's the only probabilistic evidence you have (in addition to the background knowledge). Even then, it's only useless if you need to make an absolute decision on the taxi's colour...
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