Quantitative Analysis riddle

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thorkorn
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Quantitative Analysis riddle

Postby thorkorn » Mon Oct 20, 2008 6:42 am UTC

Here is a riddle from my Quantitative Business analysis class, Enjoy!:

Estimation Process Bonus Problem: Your friend is being charged for a crime, you believe that he is 100% Innocent of the crime or P(G)= 0 where G stands for guilty.

However, the police have a test that proves that he is guilty. Furthermore this test is full proof 100% of the time or P( +I G ) = 1 and P ( + I i )= 0

What is the probability that your friend is guilty or P (G I +) = ?
In addition there is only 1 correct answer to this problem supposedly, also 'believe' may actually be 'know' because I wrote down the problem in a hurry, so that might bring up 2 answers.

I have no freaking idea what the answer is by the way...

Edit:
Well to just to throw this out there its not really a philosophical question, though I could see a possible philosophical answer because you are supposed to use
Spoiler:
decision tree analysis....


It is the probability that he is guilty, so yes he could have a 50% chance of being guilty
Last edited by thorkorn on Tue Oct 21, 2008 2:17 am UTC, edited 1 time in total.

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thc
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Re: Quantitative Analysis riddle

Postby thc » Mon Oct 20, 2008 9:41 pm UTC

Sounds more like a philosophical question to me. If you know your friend is innocent but are later proven incorrect, did you really "know" your friend to be innocent? Likewise, if you believe your friend is innocent but for the wrong reason, do you "know" your friend is innocent? IMO, no to both.

Also, why do you mention percentages? Is it possible to be 50% innocent?

Puck
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Re: Quantitative Analysis riddle

Postby Puck » Tue Oct 21, 2008 1:01 pm UTC

The only solution to the problem as stated is "you are an idiot". There is no scenario in which a completely logical person would believe something with 100% certainty when a completely foolproof test shows it to be false. (This is regardless of whether the person knows about the test value; essentially, I'm saying that there is no scenario in which a person believes with 100% certainty something that isn't true. It happens all the time, of course.)
22/7 wrote:If I could have an alternate horn that would yell "If you use your turn signal, I'll let you in" loud enough to hear inside another car, I would pay nearly any amount of money for it.

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Godskalken
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Re: Quantitative Analysis riddle

Postby Godskalken » Tue Oct 21, 2008 1:56 pm UTC

While I agree with the others stating that the solution is "one of the given probabilities is incorrect", Bayes' theorem does strangely not give a divide by zero; create a black hole that swallows the universe kind of answer.

Rather,
Spoiler:
P(G|+)=P(+|G)P(G)/P(+)=0,
since P(+) must be nonzero (otherwise he could not have gotten a positive test), P(+|G) is given as 1 and P(G) is given as 0.

Of course, stretching this further, since we have
P(i|+)=P(+|i)P(i)/P(+)=0,
there must either be another possibility in addition to G and i, or we must have P(+)=0, which again contradicts the fact that the result of the test was positive.

fyjham
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Re: Quantitative Analysis riddle

Postby fyjham » Tue Oct 21, 2008 2:07 pm UTC

Problem is both are stated as facts with probability at 100% and both contradict each-other so either:
A) There is a way for the man to be both guilty and innocent (Ignored cause I can't justify this logically unless it's a trick question).
B) The accuracy of each measure is incorrect (EG: One or both is not 100% accurate).
C) The question makes no sense.

I'm leaning towards C myself :P

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quintopia
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Re: Quantitative Analysis riddle

Postby quintopia » Tue Oct 21, 2008 6:50 pm UTC

Here's the way out. We can assume that the error rate for the test is based on testing an infinite number of people. Thus, since it has 100% accuracy, that still leaves room for a finite number of people to receive false positives.

To be more precise, lets say we have a device that always predicts guilt perfectly accurately, but some cruel hacker has set it up to return the opposite of the correct result on the 69th through 79th tests. Then, despite the fact that the probability of a random person being in this group goes to zero as the number of people tested goes to infinity, the friend could still receive a false result.

Thus, all you need to do is ask the cops for the error rate on some finite number of people, for instance the projected number of people the device will be used on in the course of its life. This should be positive, and so no contradiction.

Puck
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Re: Quantitative Analysis riddle

Postby Puck » Tue Oct 21, 2008 7:23 pm UTC

"I have developed a perfect lie detector! All we need to do to certify it is to test it on an infinite number of people!"

"Um... is that in the budget?"
22/7 wrote:If I could have an alternate horn that would yell "If you use your turn signal, I'll let you in" loud enough to hear inside another car, I would pay nearly any amount of money for it.

Christopher
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Re: Quantitative Analysis riddle

Postby Christopher » Tue Oct 21, 2008 8:21 pm UTC

He's guilty of a different crime?

/This problem sucks

Buttons
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Re: Quantitative Analysis riddle

Postby Buttons » Tue Oct 21, 2008 8:34 pm UTC

Puck wrote:"I have developed a perfect lie detector! All we need to do to certify it is to test it on an infinite number of people!"

"Um... is that in the budget?"

I hear such lie detectors are regularly used for visitors suspected of committing crimes at Hilbert's Hotel.

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jestingrabbit
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Re: Quantitative Analysis riddle

Postby jestingrabbit » Wed Oct 22, 2008 4:03 am UTC

I think Godskalken has given the correct answer here. The fact that Bayes' theorem returns sense even when its fed garbage is remarkable.
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thorkorn
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Re: Quantitative Analysis riddle

Postby thorkorn » Thu Oct 30, 2008 1:21 am UTC

The correct answer was:
Spoiler:
Indeterminable as many of you guessed

:D

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parallax
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Re: Quantitative Analysis riddle

Postby parallax » Thu Oct 30, 2008 4:32 pm UTC

since P(+) must be nonzero (otherwise he could not have gotten a positive test)


Just because something happens doesn't mean its probability is non-zero. We should take P(+)=0. Then Bayes Theorem gives P(G|+)=P(+|G)P(G)/P(+)=0/0. Or, to rewrite: P(G|+)P(+)=P(+|G)P(G)=0. This is satisfied for any probability P(G).

The question is: are we sure that he is innocent, or only almost sure, that is, sure with probability 1?
Cake and grief counseling will be available at the conclusion of the test.

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jestingrabbit
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Re: Quantitative Analysis riddle

Postby jestingrabbit » Thu Oct 30, 2008 4:51 pm UTC

parallax wrote:
since P(+) must be nonzero (otherwise he could not have gotten a positive test)


Just because something happens doesn't mean its probability is non-zero. We should take P(+)=0.


I've gotta speak against this. Sure, in an infinite, lebesgue measure space what you are saying is true, but if we're talking reality, then its false imo, at least as far as assigning probabilities using Bayes.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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Godskalken
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Re: Quantitative Analysis riddle

Postby Godskalken » Thu Oct 30, 2008 5:24 pm UTC

parallax wrote:
since P(+) must be nonzero (otherwise he could not have gotten a positive test)


Just because something happens doesn't mean its probability is non-zero. We should take P(+)=0. Then Bayes Theorem gives P(G|+)=P(+|G)P(G)/P(+)=0/0. Or, to rewrite: P(G|+)P(+)=P(+|G)P(G)=0. This is satisfied for any probability P(G).

The question is: are we sure that he is innocent, or only almost sure, that is, sure with probability 1?

I actually originally thought of what you mentioned, but basically I agree with jestingrabbit.
If you have a continuous outcome space, then the probability of any outcome will normally be 0, and thus the fact that a specific outcome occurs does not mean that the probability of its occurance is non-zero. (However, as long as we perform a finite number of tests, no outcome will ever be repeated.)
In this case, on the other hand, there is a discrete set of possible outcomes, namely + and -. Since we can safely assume that the test is only performed a finite number of times, no outcome can be + unless P(+)>0.


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