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Tags classic puzzles , math puzzles , probability , probability puzzles

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Old 23rd April 2011, 03:28 PM   #1
Ivor the Engineer
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Very hard question in online test: all answers are wrong?

I've just taken the Bang goes the Theory Big Risk Test. All answers were multiple choice. One of the questions was as follows:

Quote:
A criminal hides in a room with 99 innocent people. You have a lie detector that correctly classifies 95% of people. You pick someone at random, wire them up to the machine, and ask them if they are a criminal. They say no, but the machine goes ‘ping’ and says the person is lying. What is the chance that you have caught the criminal?
I picked the answer nearest to the one I calculated from those available (95%, 83%, 17% and 5%, IIRC):

17%


The value I calculated was...

...the PPV = 0.01*0.95/(0.01*0.95+0.99*0.05) = 16.1%, or 16% to two sf.


Have I missed something or are all the available options wrong, albeit only by a small amount?
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Old 23rd April 2011, 03:32 PM   #2
RobDegraves
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It's 95%

The machine is 95% accurate.

The person said he's not a criminal and he is lying... therefore if the machine is accurate, he's is a criminal. It's a trick question.
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Old 23rd April 2011, 03:38 PM   #3
Beth
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Originally Posted by Ivor the Engineer View Post
I've just taken the Bang goes the Theory Big Risk Test. All answers were multiple choice. One of the questions was as follows:



I picked the answer nearest to the one I calculated from those available (95%, 83%, 17% and 5%, IIRC):

17%


The value I calculated was...

...the PPV = 0.01*0.95/(0.01*0.95+0.99*0.05) = 16.1%, or 16% to two sf.


Have I missed something or are all the available options wrong, albeit only by a small amount?
I don't think you've missed anything, you've computed the exact probability.

I think what they've done is round to integers. The test is expected to identify 6 people out of the 100 as criminals. One of them actually is the criminal. Thus, the probability is 1/6 ~ 17%. Your computation also takes into account the small chance that more or less than 6 people are identified as criminals.
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Old 23rd April 2011, 03:48 PM   #4
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I believe 0.01/(0.01*0.95+0.99*0.05) = .1695
is the correct answer.
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Old 23rd April 2011, 04:48 PM   #5
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I agree with Beth
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Old 23rd April 2011, 05:12 PM   #6
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I think you are missing false positive and false negatives. I am trying to figure out the math but my best guest would be 83
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Old 23rd April 2011, 07:05 PM   #7
69dodge
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I get 19/118 = 16.1016949...%.

So, the same as Ivor the Engineer.
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Old 23rd April 2011, 07:17 PM   #8
69dodge
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Originally Posted by RobDegraves View Post
The machine is 95% accurate.
That doesn't mean what you think it means.

It means, if the machine tests someone who is lying, with probability 95% it will say he's lying and with probability 5% it will say he's telling the truth. Likewise, if the machine tests someone who is telling the truth, with probability 95% it will say he's telling the truth and with probability 5% it will say he's lying.

Suppose you're telling the truth, and the guy running the machine knows it. Perhaps he asked you, "what color is the sky?" and you said, "blue". There is a 5% chance that the machine will misclassify you and say that you're lying. If that happens, the proper conclusion is that the machine must have messed up this time; the proper conclusion is not that you're probably lying about the sky being blue.

The prior probability of lying needs to be taken into account.
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Old 23rd April 2011, 07:33 PM   #9
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Me too. Baye's Law FTW.
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Old 23rd April 2011, 07:42 PM   #10
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Originally Posted by vwgub View Post
I think you are missing false positive and false negatives. I am trying to figure out the math but my best guest would be 83
Misread the original question I would also get 17 percent based off the question as asked.
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Old 23rd April 2011, 07:58 PM   #11
69dodge
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Originally Posted by Perpetual Student View Post
I believe 0.01/(0.01*0.95+0.99*0.05) = .1695
is the correct answer.
Can you explain how you got that?
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Old 23rd April 2011, 08:55 PM   #12
marplots
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I'd to it this way. Say you tested all 100 people. There would be 100 results. 5 would be incorrect and 95 would be correct (given in the problem). What is the probability that your criminal would be in the larger group? 95%

If he is in the larger pool, the test is giving the right answer, as it will for anyone in that group. So I'd go with 95%. I don't think the joint probability with the additional information should matter.
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Old 23rd April 2011, 08:59 PM   #13
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For those thinking the answer is 95%... its a bit more complicated than that...

http://www.badscience.net/2006/12/cr...ictive-values/

http://www.badscience.net/2009/02/da...-if-it-worked/

Last edited by Professor Yaffle; 23rd April 2011 at 09:01 PM.
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Old 23rd April 2011, 09:31 PM   #14
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Originally Posted by 69dodge View Post
Can you explain how you got that?
I'm pretty sure that's just a small error. Perpetual Student is doing what Ivor et. al. are doing, except incorrectly using 0.01, rather than 0.01*95%. It's ignoring the possibility of the detector being wrong about the liar, but only in the numerator.
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Old 23rd April 2011, 09:51 PM   #15
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OK... about the 95%, I think it has a bit to do with how you establish odds and how you read the question.

Let's look at flipping coins for a clearer example if you don't mind.

If I flip a coin once, the odds of getting heads is 50%.

If I flip a coin a hundred times, I should get heads about 50 times.

The odds of getting heads on throw 59 though... are much lower. Or, the odds of getting 99 heads is lower, etc.

However, each specific throw is still 50%, no matter what happened before.

So... look at your problem and how it is worded, specifically this part.

Quote:
You pick someone at random, wire them up to the machine, and ask them if they are a criminal. They say no, but the machine goes ‘ping’ and says the person is lying. What is the chance that you have caught the criminal?
In this specific instance... you have a 95% chance of having caught the criminal.


I hope that explains my reasoning.

Last edited by RobDegraves; 23rd April 2011 at 09:53 PM.
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Old 23rd April 2011, 09:55 PM   #16
69dodge
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Originally Posted by marplots View Post
Say you tested all 100 people. There would be 100 results. 5 would be incorrect and 95 would be correct (given in the problem).
Yes.

Originally Posted by marplots View Post
What is the probability that your criminal would be in the larger group? 95%
Yes.

Originally Posted by marplots View Post
If he is in the larger pool, the test is giving the right answer, as it will for anyone in that group.
No.

Let's say the criminal is definitely in the larger group, and so he fails the test. That doesn't mean that this person who failed the test is the criminal. Five other people besides the criminal also failed. This person is probably one of them, simply because there are more of them.
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Old 23rd April 2011, 10:03 PM   #17
69dodge
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Originally Posted by RobDegraves View Post
Let's look at flipping coins for a clearer example if you don't mind.

If I flip a coin once, the odds of getting heads is 50%.

If I flip a coin a hundred times, I should get heads about 50 times.
Yes.

Originally Posted by RobDegraves View Post
The odds of getting heads on throw 59 though... are much lower.
No, they're 50% too.

Originally Posted by RobDegraves View Post
Or, the odds of getting 99 heads is lower, etc.
Yes.

Originally Posted by RobDegraves View Post
However, each specific throw is still 50%, no matter what happened before.
Yes.

Originally Posted by RobDegraves View Post
So... look at your problem and how it is worded, specifically this part.

Quote:
You pick someone at random, wire them up to the machine, and ask them if they are a criminal. They say no, but the machine goes ‘ping’ and says the person is lying. What is the chance that you have caught the criminal?
In this specific instance... you have a 95% chance of having caught the criminal.

I hope that explains my reasoning.
I don't see the connection between this problem and your examples of the coin.
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Old 23rd April 2011, 10:19 PM   #18
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I have a 95% chance of getting caught lying to my wife, but there is only a 1% chance of that.
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Old 23rd April 2011, 10:37 PM   #19
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Originally Posted by marplots View Post
I'd to it this way. Say you tested all 100 people. There would be 100 results. 5 would be incorrect and 95 would be correct (given in the problem). What is the probability that your criminal would be in the larger group? 95%

If he is in the larger pool, the test is giving the right answer, as it will for anyone in that group. So I'd go with 95%. I don't think the joint probability with the additional information should matter.
I know very little about statistics, but I get that there is only a 16.66% chance approx. that a person failing the test is actually a criminal. If all 100 suspects are tested, approx. 94 would pass the test. We can expect the machine to go ''ping'' 6 times, once for the criminal and 5 times for innocent persons. So, if only one of those 6 selected by the machine is actually guilty, that would be 1 out of 6, or 16.66%

I have not taken into account the possibility (5 out of 100) that the criminal could pass the test. I don't know the math to do that.
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Old 23rd April 2011, 10:56 PM   #20
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Presumably all 100 people involved would deny being the criminal. So the chance that the machine correctly identified the criminal would depend on how reliable the machine is - 95%.
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Old 23rd April 2011, 11:09 PM   #21
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Presumably all 100 people involved would deny being the criminal. So the chance that the machine correctly identified the criminal would depend on how reliable the machine is - 95%.
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Old 23rd April 2011, 11:10 PM   #22
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Originally Posted by RobDegraves View Post
In this specific instance... you have a 95% chance of having caught the criminal.


I hope that explains my reasoning.
No. Let's say we changed the test slightly, instead of having 100 people in the room and one is the criminal, what if we have 1000 people in the room and one is the criminal? By your logic, that doesn't change the probability, correct?

Okay, so what if we have 100 people in the room and none are criminals? Now, you pick someone at random and the lie detector goes "ping" and says he's lying. What are the chances that he's a criminal? Clearly your knowledge about how many criminals there are in the room has an impact on the probability.
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Old 23rd April 2011, 11:17 PM   #23
69dodge
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Originally Posted by psionl0 View Post
Presumably all 100 people involved would deny being the criminal.
Yes.

Originally Posted by psionl0 View Post
So the chance that the machine correctly identified the criminal would depend on how reliable the machine is - 95%.
The question isn't: if the criminal takes the test, what's the probability that the test says he's a criminal?

The question is: if a random person takes the test, and the test says he's a criminal, what's the probability that he is the criminal?

(The answer to the first question is 95%. But we don't really care about that question, because it assumes that we already know who the criminal is. Which, of course, we don't; we're trying to find him.)
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Old 23rd April 2011, 11:20 PM   #24
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Ivor, the test gives you all the results at the end.


5.A criminal hides in a room with 99 innocent people. You have a lie detector that correctly classifies 95% of people. You pick someone at random, wire them up to the machine, and ask them if they are a criminal. They say no, but the machine goes ‘ping’ and says the person is lying. What is the chance that you have caught the criminal?
Your answer: 95%. Correct answer: 17%.


(Yeah, I got that one wrong, too. )
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Old 23rd April 2011, 11:31 PM   #25
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Since you picked the person randomly, there's a 1% chance that you have the criminal, and a 99% chance that you don't have him. Let's say you got an innocent person, but the machine still goes 'ping'. You just found one of the 5 times in a hundred that the machine is wrong. Similarly, if you do have him then there's a 5% chance that the machine won't 'ping'.

Compare the chances - 95% of the 1% chance that you do have him, but 5% of the 99% chance that you don't. It's much more likely that you got the wrong answer (false detection) than the correct one.
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Old 23rd April 2011, 11:52 PM   #26
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Another way of looking at it. You have a signal whose strength is 1% of full scale, but your detector has a noise level of 5%. Your signal is buried in the noise!
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Old 24th April 2011, 12:46 AM   #27
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Originally Posted by psionl0 View Post
Presumably all 100 people involved would deny being the criminal. So the chance that the machine correctly identified the criminal would depend on how reliable the machine is - 95%.
No, because there are false positive. The easiest way to do that is to use bigger number instead : they are not testing 100 person but 300 million, the whole USA population. How many will be "pinged" as being criminal ? 15 million. But there is only 1 criminal.

So now you have 15 million person and one of them is the criminal, but you have no further test at your disposition.

Still thinking that the probability of detecting the criminal is 95% ? It is clearly not.

By the way this is why testing a whole population with even an ultra mega super precise "positive test" don't work. Due to the amount of false positive.
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Old 24th April 2011, 01:23 AM   #28
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Originally Posted by psionl0 View Post
Presumably all 100 people involved would deny being the criminal. So the chance that the machine correctly identified the criminal would depend on how reliable the machine is - 95%.
The machine will identify approximately 6 criminals but approximately 5 of these identifications will be wrong.
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Old 24th April 2011, 01:24 AM   #29
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Originally Posted by 69dodge View Post
The question isn't: if the criminal takes the test, what's the probability that the test says he's a criminal?

The question is: if a random person takes the test, and the test says he's a criminal, what's the probability that he is the criminal?
OK got the question wrong. This actually turns out to be a standard style of question that you might find in a mathematics text book.

If the person is a criminal then there is a 95% chance that the machine will call him a liar. There is a 1% chance that the person is a criminal so there is a 95% x 1% = 0.95% that the person is a criminal and the machine calls him a liar.

If the person is not a criminal then there is a 5% chance that the machine will call him a liar. There is a 99% that the person is not a criminal so there is a 5% x 99% = 4.95% chance that the person is not a criminal and the machine calls him a liar.

So the probability that the machine will say "liar" is 4.95% + 0.95% = 5.9%.

According to standard conditional probability forumae, the probability that a person is a liar (criminal) given that the machine said "liar" is equal to the probability that the person is a criminal and the machine said "liar" divided by the probability that the machine said "liar".

ie: 0.95% / 5.9% = 16.61017% the same as the OP.

Last edited by psionl0; 24th April 2011 at 01:34 AM.
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Old 24th April 2011, 01:45 AM   #30
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Originally Posted by yomero View Post
I know very little about statistics, but I get that there is only a 16.66% chance approx. that a person failing the test is actually a criminal. If all 100 suspects are tested, approx. 94 would pass the test. We can expect the machine to go ''ping'' 6 times, once for the criminal and 5 times for innocent persons. So, if only one of those 6 selected by the machine is actually guilty, that would be 1 out of 6, or 16.66%

I have not taken into account the possibility (5 out of 100) that the criminal could pass the test. I don't know the math to do that.

You've pretty much got it; except approximating to the nearest integer obscures the math a bit. (If we only run the test once, we will get integers; it might help to think of running the test more than once so we get statistical averages instead to work with.)

Each innocent has a 5% change of pinging. So on average, 99 x .05 = 4.95 pings from the innocents.

Each criminal has a 95% chance of pinging. So on average, 1 x .95 = .95 pings from the criminal.

Total pings (per 100 in the room) on average, 4.95 + .95 = 5.90.

Chance a ping is the criminal: average criminal pings / average total pings = .95 / 5.90 = 16.1%.
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Old 24th April 2011, 05:04 AM   #31
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For a more real life related problem. The government wants to detect any spies in a large organisation. There would be thousands of loyal workers to every spy. A lie detector would have to be very reliable, much better than 95%, to be worth anything. To make matters even worse for the government the spy may have received training on how to pass the test making a false negative more likely than a false positive.
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Old 24th April 2011, 05:59 AM   #32
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Pretty good tests, I ended up being a social risk taker, with mid-low on everything else.

I got all "understanding risks" questions right except I miscalculated the odds of the dice coming up 6, apparently.
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Old 24th April 2011, 07:03 AM   #33
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Why is everyone so confused about this simple problem:

In a room of 100 people only one of whom is a criminal, a machine which has a 95% accuracy indicates that a randomly selected person is the criminal. What's the probability that the machine has actually picked the criminal.

P = Ptrue positive/(Ptrue postiive + Pfalse positive)
Ptrue postiive = 1/100 x 95/100
Pfalse positive = 99/100 x 5/100
P = .161016949 or 16.1016949%
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Old 24th April 2011, 07:14 AM   #34
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I just thought, "OK, in 100 people there will be about 5 false positives and 1 true positive. Therefore, the odds that any positive is a true positive is about 100/6, or 16-17%."
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Old 24th April 2011, 07:16 AM   #35
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Originally Posted by TubbaBlubba View Post
I just thought, "OK, in 100 people there will be about 5 false positives and 1 true positive. Therefore, the odds that any positive is a true positive is about 100/6, or 16-17%."
Me too: "close enough for government work".
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Old 24th April 2011, 07:33 AM   #36
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Originally Posted by TubbaBlubba View Post
I just thought, "OK, in 100 people there will be about 5 false positives and 1 true positive. Therefore, the odds that any positive is a true positive is about 100/6, or 16-17%."
Originally Posted by Professor Yaffle View Post
Me too: "close enough for government work".
Glad it wasn't just me that looked at it like that. I wasn't going to say that was how I did it only to be shown that I should stick to counting on my fingers!
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Old 24th April 2011, 08:07 AM   #37
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Originally Posted by TubbaBlubba View Post
Pretty good tests, I ended up being a social risk taker, with mid-low on everything else.
Yes; Social was my biggest risk area, and that was Low.

Quote:
I got all "understanding risks" questions right except I miscalculated the odds of the dice coming up 6, apparently.
Same here; maybe the phrasing threw me.
I know it's a one in six chance of a particular number coming up, but does that really mean that on average it takes six throws to get that particular number?


ETA:I got 85% on the Judging risks, doing much better on the non-lethal than lethal risks (the former were nearly all bang-on, the latter were all over the place.
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Old 24th April 2011, 08:36 AM   #38
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The dice one threw some other people apparently (found some threads discussing it).

http://www.physicsforums.com/showthread.php?p=3261226
http://www.wordmagazine.co.uk/content/take-a-chance-me

ETA - if it were drawing from 6 numbers without replacement, I think the average would be 3.5, but because you can always throw the same number more than once, the answer is 6. I'll let someone more mathsy than me explain it better.

Last edited by Professor Yaffle; 24th April 2011 at 08:43 AM.
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Old 24th April 2011, 09:25 AM   #39
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The fact that there are a significant number of false positives, and negatives, in a lie detector test is one of the reasons why that lie detectors are not usable in a court of law. When used on a group of random people the finding of guilt is much more likely to be wrong than right if the false answers are anywhere near the percentage given in this example. When it comes to a crime there is always a much much larger pool of innocent people than guilty people, so an indicator of guilty for anyone in that random group is probably wrong. It could even be probably wrong if you had a large enough pool of suspects.

When they first started testing for the AIDS virus the early tests had a high number of false positives. If you received a positive test result on one of those early tests it was not necessarily the end of the world. If you were a truly random subject the odds were that you did not have the disease. But people usually had a reason to be tested so the number of actual false positives was lower than that random number.
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Old 24th April 2011, 10:30 AM   #40
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Originally Posted by marplots View Post
I'd to it this way. Say you tested all 100 people. There would be 100 results. 5 would be incorrect and 95 would be correct (given in the problem). What is the probability that your criminal would be in the larger group? 95%

If he is in the larger pool, the test is giving the right answer, as it will for anyone in that group. So I'd go with 95%. I don't think the joint probability with the additional information should matter.
But it does matter, because you're only given one test result, and that result is positive. Any negatives have already been discarded. Your "population" that you are sampling from is all the positive results (accurate or not), NOT all the people. And because the probability of being a criminal is smaller than the probability of getting a false result, most of the positive results are false positives.
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