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Math Help - Is this conditional Probability?

  1. #1
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    Is this conditional Probability?

    Here is the crux of the problem:

    A man gets hit-and-run by a taxi. An eye-witness says the taxi is orange. 85% of the taxis in town are crimson and 15% are orange. The accident occurs at night and the probability that the eye-witness mistook either for the other is 0.2. The man sues the ORANGE company. But, the judge will decide in favor of the man only if the evidence points to a probability of more than 0.7. How does the judge find? Why?

    I'm not sure whether this problem requires conditional probability. The way I see it is this:
    The probability that witness is confused = 0.2
    The probability that the witness is wrong = 0.2 x 0.85 = 0.17

    Since the Judge requires 70% probability, she will find in favor of the man, irrespective of what the proportion of taxis is in the city. I say this as the maximum probability that the eye-witness is WRONG is 0.2. Hence, the judge can be AT LEAST 80% sure about the orange company being guilty.

    Does my argument sound logical? Is there something I'm ignoring? If I'm right, that means there's no point in including details about the proportion of taxis. Thats whats bugging me. Would Bayes' theorem apply to this? In that case, am I wrong? PLEASE REPLY!
    Last edited by indian; February 21st 2008 at 09:12 PM.
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    Quote Originally Posted by indian View Post
    Here is the crux of the problem:

    A man gets hit-and-run by a taxi. An eye-witness says the taxi is orange. 85% of the taxis in town are crimson and 15% are red. The accident occurs at night and the probability that the eye-witness mistook either for the other is 0.2. The man sues the ORANGE company. But, the judge will decide in favor of the man only if the evidence points to a probability of more than 0.7. How does the judge find? Why?

    I'm not sure whether this problem requires conditional probability. The way I see it is this:
    The probability that witness is confused = 0.2
    The probability that the witness is wrong = 0.2 x 0.85 = 0.17

    Since the Judge requires 70% probability, she will find in favor of the man, irrespective of what the proportion of taxis is in the city. I say this as the maximum probability that the eye-witness is WRONG is 0.2. Hence, the judge can be AT LEAST 80% sure about the orange company being guilty.

    Does my argument sound logical? Is there something I'm ignoring? If I'm right, that means there's no point in including details about the proportion of taxis. Thats whats bugging me. Would Bayes' theorem apply to this? In that case, am I wrong? PLEASE REPLY!
    If taxis are either crimson or red, what's the deal with orange .......??

    Be that as it may, conditional probability will be involved.
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    Omg, I'm so stupid. Sorry about that, edited it.

    But wait, HOW will conditional probability be involved? Is my solution right as well, then? Is this about Bayes' theorem?
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    Quote Originally Posted by indian View Post
    Here is the crux of the problem:

    A man gets hit-and-run by a taxi. An eye-witness says the taxi is orange. 85% of the taxis in town are crimson and 15% are orange. The accident occurs at night and the probability that the eye-witness mistook either for the other is 0.2. The man sues the ORANGE company. But, the judge will decide in favor of the man only if the evidence points to a probability of more than 0.7. How does the judge find? Why?

    I'm not sure whether this problem requires conditional probability. The way I see it is this:
    The probability that witness is confused = 0.2
    The probability that the witness is wrong = 0.2 x 0.85 = 0.17

    Since the Judge requires 70% probability, she will find in favor of the man, irrespective of what the proportion of taxis is in the city. I say this as the maximum probability that the eye-witness is WRONG is 0.2. Hence, the judge can be AT LEAST 80% sure about the orange company being guilty.

    Mr F says: By your argument, if 99.999% of taxis in town are crimson and 0.001% are orange then the probability that the eyewitness is wrong is about 0.2. In other words, it's highly likely that the witness saw an orange taxi!! Do you see how illogical that is? If you don't, then consider the extreme case of 100% of taxis in town are crimson and 0% are orange ...... See it now?

    Does my argument sound logical? Is there something I'm ignoring? If I'm right, that means there's no point in including details about the proportion of taxis. Thats whats bugging me. Would Bayes' theorem apply to this? In that case, am I wrong? PLEASE REPLY!
    ..
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    Quote Originally Posted by indian View Post
    Here is the crux of the problem:

    A man gets hit-and-run by a taxi. An eye-witness says the taxi is orange. 85% of the taxis in town are crimson and 15% are orange. The accident occurs at night and the probability that the eye-witness mistook either for the other is 0.2. The man sues the ORANGE company. But, the judge will decide in favor of the man only if the evidence points to a probability of more than 0.7. How does the judge find? Why?

    I'm not sure whether this problem requires conditional probability. The way I see it is this:
    The probability that witness is confused = 0.2
    The probability that the witness is wrong = 0.2 x 0.85 = 0.17

    Since the Judge requires 70% probability, she will find in favor of the man, irrespective of what the proportion of taxis is in the city. I say this as the maximum probability that the eye-witness is WRONG is 0.2. Hence, the judge can be AT LEAST 80% sure about the orange company being guilty.

    Does my argument sound logical? Is there something I'm ignoring? If I'm right, that means there's no point in including details about the proportion of taxis. Thats whats bugging me. Would Bayes' theorem apply to this? In that case, am I wrong? PLEASE REPLY!
    You're completely on the wrong track, I'm sorry to say. The given data is:

    Pr(Taxi is Crimson) = 0.85.
    Pr(Taxi is Orange) = 0.15.
    Pr(Eyewitness sees crimson taxi | taxi is orange) = 0.2
    Pr(Eyewitness sees orange taxi | taxi is crimson) = 0.2. Therefore Pr(Eyewitness sees orange taxi | taxi is orange) = 0.8.

    Then:

    Pr(Taxi is orange | Eyewitness sees orange taxi) Pr(Eyewitness sees orange taxi) = Pr(Eyewitness sees orange taxi | taxi is orange) Pr(Taxi is orange)

    => Pr(Taxi is orange | Eyewitness sees orange taxi) (1) = (0.8) (0.15) = 0.12.

    So there's only a 12% chance that the taxi the eyewitness saw is orange.

    The defence rests, Your Honour.

    Postscript: Since the Judge required a 70% probability that the taxi the eyewitness saw was orange, the judge found against the man.
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    Quote Originally Posted by mr fantastic View Post
    You're completely on the wrong track, I'm sorry to say. The given data is:

    Pr(Taxi is Crimson) = 0.85.
    Pr(Taxi is Orange) = 0.15.
    Pr(Eyewitness sees crimson taxi | taxi is orange) = 0.2
    Pr(Eyewitness sees orange taxi | taxi is crimson) = 0.2. Therefore Pr(Eyewitness sees orange taxi | taxi is orange) = 0.8.

    Then:

    Pr(Taxi is orange | Eyewitness sees orange taxi) Pr(Eyewitness sees orange taxi) = Pr(Eyewitness sees orange taxi | taxi is orange) Pr(Taxi is orange)

    => Pr(Taxi is orange | Eyewitness sees orange taxi) (1) = (0.8) (0.15) = 0.12.

    So there's only a 12% chance that the taxi the eyewitness saw is orange.

    The defence rests, Your Honour.

    Postscript: Since the Judge required a 70% probability that the taxi the eyewitness saw was orange, the judge found against the man.
    Post Postscript:

    If this were a criminal case in the UK a probabilistic argument would be inadmissible.

    Apparently beyond resonable doubt does not allow a Bayesian argument because the
    jury on top of the Clapham omnibus would not understand it.

    RonL
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    Quote Originally Posted by CaptainBlack View Post
    Post Postscript:

    If this were a criminal case in the UK a probabilistic argument would be inadmissible.

    Apparently beyond resonable doubt does not allow a Bayesian argument because the
    jury on top of the Clapham omnibus would not understand it.

    RonL
    Indeed. The famous case of R v Adams . If it wasn't so pathetic it would be laughable.

    Some references of possible interest:

    http://www.stat.auckland.ac.nz/~iase.../2/Topic4q.pdf

    http://www.maths.sussex.ac.uk/Staff/.../FSLPnotes.pdf

    And there's been a couple of articles of interest in new Scientist:

    16 April 1994 (pp 12 - 13)

    8 June 1996 (p7)

    13 December 1997 (pp 18 - 19)
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