A taxi was involved in a hit and run accident at night. Two taxi companies, the green and the blue, operate in the city. 85% percent of the taxis are green and 15% are blue. A witness identified the taxi as blue.The witness identifies the correct color 80% of the time and fails 20% of the time. What is the probability that the taxi was blue?
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Taxi problem
- Thread starter BobRodes
- Start date
CajunCenturion
Programmer
29%
The probability that the taxis was actually blue and properly identified as blue (.15 * .80 = .12) plus the probability that the taxis was actually green but mis-identified as blue (.85 * .20 = .17)
The probability that the taxis was actually blue and properly identified as blue (.15 * .80 = .12) plus the probability that the taxis was actually green but mis-identified as blue (.85 * .20 = .17)
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[hide]Ooh - ohh, I seem to recall seeing this one before. Um, let me think ... yes, we need a table ...[tt]
Green Blue
Unwitnessed odds: 85% 15%
Witness claims:
Green taxi 80% of 85% = 68% 20% of 15% = 3% => 71% of time witness says green
Blue taxi 20% of 85% = 17% 80% of 15% = 12% => 29% of time witness says blue
We can now calculate the following
1. odds witness correctly identified green as green: 68/71 = 95.77%
2. odds witness misidentified green as blue: 3/71 = 4.23%
3. odds witness correctly identified blue as blue: 12/29 = 41.38%
4. odds witness misidentified blue as green: 17/29 = 58.62%[/tt]
And from this we can see that the answer is line 3 above, 41.38%[/hide]
Green Blue
Unwitnessed odds: 85% 15%
Witness claims:
Green taxi 80% of 85% = 68% 20% of 15% = 3% => 71% of time witness says green
Blue taxi 20% of 85% = 17% 80% of 15% = 12% => 29% of time witness says blue
We can now calculate the following
1. odds witness correctly identified green as green: 68/71 = 95.77%
2. odds witness misidentified green as blue: 3/71 = 4.23%
3. odds witness correctly identified blue as blue: 12/29 = 41.38%
4. odds witness misidentified blue as green: 17/29 = 58.62%[/tt]
And from this we can see that the answer is line 3 above, 41.38%[/hide]
trying to use ststistics on the above question is flawd because you are working with a sample size of 1
if the witness was being asked to identify the collory of every taxi in the fleet then che chanses of a taxi identified as blue would indead match on of the suggestions above (to be honest i havent tried to work out which but strongm's figures of 71 29 for the prediction look correct)
The point is the witnes is being asked to identift 1 taxi & it has already been defined that his accuracy is 80%
if the witness was being asked to identify the collory of every taxi in the fleet then che chanses of a taxi identified as blue would indead match on of the suggestions above (to be honest i havent tried to work out which but strongm's figures of 71 29 for the prediction look correct)
The point is the witnes is being asked to identift 1 taxi & it has already been defined that his accuracy is 80%
[hidden]
Further analasis Strongms figures are correct but the data is badly formed because there is only 1 event which is statisticaly inconclusive & has a High Margin for error
likewise accepting the witnesses 80% is also badly formed because again there is only 1 event resulting in a similar large margin of error.
My real world conclusion is:
1) Evidence from s single witness should not be given a high credibility without supporting evidence
[/hidden]
Further analasis Strongms figures are correct but the data is badly formed because there is only 1 event which is statisticaly inconclusive & has a High Margin for error
likewise accepting the witnesses 80% is also badly formed because again there is only 1 event resulting in a similar large margin of error.
My real world conclusion is:
1) Evidence from s single witness should not be given a high credibility without supporting evidence
[/hidden]
@ IPGuru - Im seeing a pattern here to your logic processes. You seem to think that single instances have relevance on probability.
Probability is the likelyhood of a single event being true.
Margins of error do not apply when the sampling data is taken as 100% accurate.
@Strongm - Kudo's. I got to stage 1 for your solution but I was stuck on how to relate them to one another. Brilliant.
@CajunCenturion - Your answer would be correct if the question was "what is the probability the witness will say blue" As you know, that is not the question.
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Occam's Razor - All things being equal, the simplest solution is the right one.
Probability is the likelyhood of a single event being true.
Margins of error do not apply when the sampling data is taken as 100% accurate.
@Strongm - Kudo's. I got to stage 1 for your solution but I was stuck on how to relate them to one another. Brilliant.
@CajunCenturion - Your answer would be correct if the question was "what is the probability the witness will say blue" As you know, that is not the question.
*******************************************************
Occam's Razor - All things being equal, the simplest solution is the right one.
CajunCenturion
Programmer
You are absolutely correct, kwbMitel. I answered the wrong question.
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Good Luck
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As a circle of light increases so does the circumference of darkness around it. - Albert Einstein
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Good Luck
To get the most from your Tek-Tips experience, please read
FAQ181-2886
As a circle of light increases so does the circumference of darkness around it. - Albert Einstein
G = Percent of green taxis
B = Percent of blue taxis
R = Percent correct
W = Percent incorrect
Probability of blue = B*R*100/(B*R + G*W) = 41.3%
B = Percent of blue taxis
R = Percent correct
W = Percent incorrect
Probability of blue = B*R*100/(B*R + G*W) = 41.3%
mmerlinn
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Prattaratt
Technical User
The Key phrase is "The witness identifies the correct color 80% of the time and fails 20% of the time." If the Witness which identifies the color correctly 80% of the time identifies it as a blue taxi, then there is an 80% chance it is correct, and therefore a blue taxi. Color distribution of the taxis is irrelevent and thrown as a red herring, and should be disregarded
@ CajunCenturion
So this is how it feels to be correct. Somehow I thought it would be more satisfying. I feel like bookmarking it anyway just in case this is the one and only time those words are ever written.
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Occam's Razor - All things being equal, the simplest solution is the right one.
So this is how it feels to be correct. Somehow I thought it would be more satisfying. I feel like bookmarking it anyway just in case this is the one and only time those words are ever written.
*******************************************************
Occam's Razor - All things being equal, the simplest solution is the right one.
Prattaratt i thought colour distribution was irelavant until are started to explain why then realised my error
The car is either 1 of 15% blue cars or 1 of 85% green cars
using the 80% accuracy of the witness he would sucsessfully identify 12% blue. however with the same 80% he would have incorrectly identified 17% of the green cars as blue
or to put it another way if there were 100 cars he has een 1 car from a choice of 17 green & 12 blue
so the ratio is 17-12 that he has selected a wrong car.
interestingly the results is different if he thought it was a green car
in those cases it would have been 68:3 that he has selected the correct car.
The car is either 1 of 15% blue cars or 1 of 85% green cars
using the 80% accuracy of the witness he would sucsessfully identify 12% blue. however with the same 80% he would have incorrectly identified 17% of the green cars as blue
or to put it another way if there were 100 cars he has een 1 car from a choice of 17 green & 12 blue
so the ratio is 17-12 that he has selected a wrong car.
interestingly the results is different if he thought it was a green car
in those cases it would have been 68:3 that he has selected the correct car.
Strongm - Re:Color distribution of the taxis is irrelevent
My understanding of the word irrelevent is that the accociated values can be removed or modified without affecting the outcome.
Either my understanding of the word irrelevent is incorrect or your usage is. Which?
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Occam's Razor - All things being equal, the simplest solution is the right one.
My understanding of the word irrelevent is that the accociated values can be removed or modified without affecting the outcome.
Either my understanding of the word irrelevent is incorrect or your usage is. Which?
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Occam's Razor - All things being equal, the simplest solution is the right one.
You perhaps need to read what I said more carefully; I was disagreeing with the assertion that the color distribution of the taxis was irrelevant. i.e. it is relevant - as my table makes pretty clear.
So my usage is correct and concurs with your understanding of the word, but your comprehension of what I said is incorrect.
So my usage is correct and concurs with your understanding of the word, but your comprehension of what I said is incorrect.
Strongm - Ahhhh, the self contradiction was evident but the context was not. Thanks for the clarification. I was reading the statement as yours, not a quote of someone else's. nuff said.
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Occam's Razor - All things being equal, the simplest solution is the right one.
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Occam's Razor - All things being equal, the simplest solution is the right one.
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