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Tags classic puzzles , math puzzles , probability , probability puzzles

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Old 24th April 2011, 11:01 AM   #41
TubbaBlubba
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Originally Posted by zooterkin View Post
Yes; Social was my biggest risk area, and that was Low.



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.
Not that one, the one whether it was more likely for a coin to come up 6 heads in a row or (I think, I may have read it wrong) for two dice to both come up with 6.

I calculated the coin odds to be about 1.6% (just 0.5^6), and figured that the dice must be 1 in 36, which is about 2.5-3%. So I'm not sure what went wrong.

Wait, maybe I just saw wrong and it was 5 heads, not 6? That might be it.
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Old 24th April 2011, 11:06 AM   #42
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Originally Posted by TubbaBlubba View Post
Wait, maybe I just saw wrong and it was 5 heads, not 6? That might be it.
I'm not taking the test because I don't want to register, so I can't see the question. But maybe it asked for the chance that the coin came up the same face 6 times in a row. Which would be the sum of the chance of coming up with 6 heads and the chance of coming up with 6 tails (which would be equal to the chance of coming up with 5 heads).
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Old 24th April 2011, 11:13 AM   #43
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Originally Posted by Ziggurat View Post
I'm not taking the test because I don't want to register, so I can't see the question. But maybe it asked for the chance that the coin came up the same face 6 times in a row. Which would be the sum of the chance of coming up with 6 heads and the chance of coming up with 6 tails (which would be equal to the chance of coming up with 5 heads).
That's possible, yeah.
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Old 24th April 2011, 01:05 PM   #44
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Knowing nothing about math or probability, I calculated this at something less than 20%. I figured that the machine would ping six times out of 100 people, and one of those would be the criminal. 1/6 is 17%.

However, this problem made my head ache so badly that I needed a CAT scan.
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Old 24th April 2011, 01:12 PM   #45
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Originally Posted by TubbaBlubba View Post
That's possible, yeah.
"Which is more likely to occur: Throwing a double-six with two fair dice or flipping a fair coin 5 times and it coming up heads every time?
Your answer: Throwing a double six with two fair dice. Correct answer: Flipping a fair coin 5 times and it coming "

Well, that explains it, it was 5 times, not 6.
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Old 24th April 2011, 03:23 PM   #46
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Originally Posted by Ziggurat View Post
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.
OK, I'm convinced. (Let it not be said that no one changes their opinions in the JREF forum!)

So, how many times should I repeat the test with that person to gain enough confidence to arrest them? (Take each test as a new "clean" shot with 95% probability of being accurate -- false positives and negatives occur randomly.)

If I understand my lessons, simply repeating once should either clear or convict.
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Old 24th April 2011, 03:32 PM   #47
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If you run through all the people twice, can you multiply the error chances together?

'Cause then there would be a .0025% chance of a double false positive or negative.
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Old 24th April 2011, 03:51 PM   #48
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The question in the OP says the machine “correctly classifies 95% of people”. I would assume that means the machine is right 95% of the time and is wrong 5% of the time. So the answer would be 16.10169%.

But if one were to read it as meaning that the machine always pings when someone tells a lie but also pings 5% of the time when someone tells the truth, then the answer is 16.80672%.
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Old 24th April 2011, 04:05 PM   #49
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Originally Posted by marplots View Post
OK, I'm convinced. (Let it not be said that no one changes their opinions in the JREF forum!)

So, how many times should I repeat the test with that person to gain enough confidence to arrest them? (Take each test as a new "clean" shot with 95% probability of being accurate -- false positives and negatives occur randomly.)

If I understand my lessons, simply repeating once should either clear or convict.
Yeah, just once or a few times will lower the probability to absurdity.
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Old 24th April 2011, 04:33 PM   #50
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Originally Posted by TubbaBlubba View Post
Yeah, just once or a few times will lower the probability to absurdity.
Assuming no correlation between trials. If there is some correlation, then the answer depends on the strength of that correlation. For example, if the correlation is 100% (ie, whatever causes the error is always present with the subject), then the probability won't change at all with repetition.
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Old 24th April 2011, 05:04 PM   #51
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Originally Posted by Dilb View Post
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.
Yes.
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Old 24th April 2011, 05:21 PM   #52
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Excuse my bartender logic, but I gathered that the question stated that only one person was tested. That one person came up lying, so the start is a probability of 1% that you picked the guilty party to test, and a 95% chance that the test is accurate, rounded to the nearest whole number you still get a 1% chance of that occurring in the first place!

They make it easy by having 100 people, the question is bogus. Percentages actually get smaller with the error of the device taken into consideration. Nowhere did I read they tested everybody.
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Old 24th April 2011, 05:33 PM   #53
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Originally Posted by Illustronic View Post
Excuse my bartender logic, but I gathered that the question stated that only one person was tested. That one person came up lying, so the start is a probability of 1% that you picked the guilty party to test, and a 95% chance that the test is accurate, rounded to the nearest whole number you still get a 1% chance of that occurring in the first place!

They make it easy by having 100 people, the question is bogus. Percentages actually get smaller with the error of the device taken into consideration. Nowhere did I read they tested everybody.
No, the fact that the result was positive gives us information. It is much more likely for the guilty to yield a positive than for an innocent, making the likelihood that we've got the guilty one higher than 1%, which would be the probability if we had no information.
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Old 24th April 2011, 05:48 PM   #54
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Originally Posted by Illustronic View Post
Excuse my bartender logic, but I gathered that the question stated that only one person was tested. That one person came up lying, so the start is a probability of 1% that you picked the guilty party to test, and a 95% chance that the test is accurate, rounded to the nearest whole number you still get a 1% chance of that occurring in the first place!
Uh, yes, if I am understanding you correctly: You have a chance that is almost 1% of finding the one guilty person and identifying him correctly with the lie-detector.

But that is what you know before actually turning the lie detector on.

In the puzzle, you are in a situation where the lie detector already beeped (which it will not do very often!) and you are looking at a somewhat different situation: You know you are not in one of the many situations where the lie detector remains silent.

It did beep. So you either have the very rare situation of having found who you were looking for, or the still quite rare situation of the lie detector giving a false positive. So we are actually working out how likely it is that one rare events happens as opposed to another rare event. (Again, in a situation where the most likely of all events failed to occur in the first place.)

Quote:
They make it easy by having 100 people, the question is bogus.
How is the question bogus?

Quote:
Percentages actually get smaller with the error of the device taken into consideration. Nowhere did I read they tested everybody.
What percentages get smaller?

No, they didn't test everybody. We only know that if they did, there would be a number of false alarms. (We don't know how many exactly. It's a shortcut to say that a 95% chance will translate into exactly 5% of false alarms in any given situation. But since we are only interested in the chances here, it's permissible to do that: The guy we found might still be guilty, even though the chance he is is rather small.)
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Old 24th April 2011, 06:02 PM   #55
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Originally Posted by Illustronic View Post
Excuse my bartender logic, but I gathered that the question stated that only one person was tested. That one person came up lying, so the start is a probability of 1% that you picked the guilty party to test, and a 95% chance that the test is accurate, rounded to the nearest whole number you still get a 1% chance of that occurring in the first place!
What if the test were 100% accurate? (That is, it never makes a mistake?) Your reasoning would suggest that, if it says the person is lying, it still has a 99% chance of having made a mistake, which is self-contradictory.

The way to interpret "the test is 95% accurate" is that, before a particular test is done, the probability is 95% that it will give the right answer, and the probability is 5% that it will give the wrong answer. However, after the test is done, the probability that it did give the right answer depends on what answer it actually gave.
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Old 24th April 2011, 06:26 PM   #56
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Originally Posted by marplots View Post
If I understand my lessons, simply repeating once should either clear or convict.
Why repeat the test? You already have evidence that the suspect is lying, so the next step to take him down to the police station and beat a confession out of him!
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Old 24th April 2011, 08:34 PM   #57
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Just in case anyone still does not get it (this is basically the same calculation BillyJoe did).

Bayes' Law

P(A|B) = (P(B|A) x P(A)) / (P(B))

P(A|B) reads as "the probability of A given B."

In our case A is whether or not the subject is a criminal, B is the results of the lie detector test.

P(B|A) = 0.95 because the test is 95% accurate. If he is a criminal the probability of getting that result is 0.95.

P(A) = 0.01 because there is only 1 criminal out of 100 people

P(B) is the total probability of getting that lie detector result regardless of criminal state. If you tested everyone in the room how many hits would you get.

P(B) = 0.95 x 0.01 (the one criminal) + 0.05 x 0.99 (for the innocent people) = 0.059

(.95 * .01) / .059 = 0.161016949

The people who calculated 1/6 = .16666666667 got very close without knowing the theory. Kudos to them.
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Old 24th April 2011, 09:56 PM   #58
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Originally Posted by ThunderChunky View Post
In our case A is whether or not the subject is a criminal, B is the results of the lie detector test.
That could be clearer. I'd have said: A means "the subject is a criminal", and B means "the lie detector reports that the subject is a criminal".

Otherwise, I agree.
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Old 24th April 2011, 11:04 PM   #59
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Originally Posted by BillyJoe View Post
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%

I still don't see why anyone is confused about this simple problem.
Well, maybe I can:

I think the problem is that those who are confused, need to concentrate on why the correct solution is correct not on why their solution is wrong. Once you see why the correct solution is correct, the reason why your solution is wrong will be obvious.

Is that clear?
...or have I confused you?
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Old 24th April 2011, 11:07 PM   #60
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...oops, I didn't see ThunderChucky's and 69Dodge's responses
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Old 24th April 2011, 11:20 PM   #61
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An interesting dilemma, I guess this discussion will end with a difference as big as 79% between the opinions of some discussers. You wouldn´t expect that from a math discussion, would you?

Some people want to think straightforward: the test is 95% reliable, so the probability of the reported test result being correct is 95%.

Others think that the 95% probability of the test result being correct somehow _changes_ when we think about how many different people could possibly be included within the 5% of false positives. This viewpoint does not hold water, methinks: the probability of the test being correct is given as 95%, so when you have as many suspects as you ever wish, and you do the test 100 times, 10,000 times or 1000,000 times, you still get the correct indication with 95% of takes and incorrect indication with 5% of takes.

Thus the probability of the indication being correct is 95%.
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Old 24th April 2011, 11:31 PM   #62
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Originally Posted by JJM 777 View Post
An interesting dilemma, I guess this discussion will end with a difference as big as 79% between the opinions of some discussers. You wouldn´t expect that from a math discussion, would you?

Some people want to think straightforward: the test is 95% reliable, so the probability of the reported test result being correct is 95%.

Others think that the 95% probability of the test result being correct somehow _changes_ when we think about how many different people could possibly be included within the 5% of false positives. This viewpoint does not hold water, methinks: the probability of the test being correct is given as 95%, so when you have as many suspects as you ever wish, and you do the test 100 times, 10,000 times or 1000,000 times, you still get the correct indication with 95% of takes and incorrect indication with 5% of takes.

Thus the probability of the indication being correct is 95%.
Really? So, if we had a population of, say a million people, with only one guilty person and the single one you happened to choose at random "rang the bell," you would conclude with a 95% certainty that you had the guilty person and that there is only a 5% chance that you chose an innocent person who happened to ring the bell? -- out of a million people? It makes no difference how many people we start with? It will always be 95%? Think man!
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Old 24th April 2011, 11:48 PM   #63
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Originally Posted by Perpetual Student View Post
Really? So, if we had a population of, say a million people, with only one guilty person and the single one you happened to choose at random "rang the bell," you would conclude with a 95% certainty that you had the guilty person and that there is only a 5% chance that you chose an innocent person who happened to ring the bell? -- out of a million people? It makes no difference how many people we start with? It will always be 95%? Think man!
Or, as I said, go a step further: what if there are no guilty people?

Imagine, for instance, a test that is 95% accurate at identifying martians. Test someone at random, and the test goes "ping". What are the chances that the person is a martian? By JJM's logic, 95%.
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Old 24th April 2011, 11:55 PM   #64
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Originally Posted by Roger Ramjets View Post
Why repeat the test? You already have evidence that the suspect is lying, so the next step to take him down to the police station and beat a confession out of him!
That's not the way it works. First you decide who is guilty based on gut feeling, then you tell them they failed the lie detector, then you beat the confession out of them.
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Old 25th April 2011, 12:04 AM   #65
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On second thought, I made an analogy for myself, and realized what exactly is the logical error in this type of evidence.

The test indicates that it is 95% probable that the guilty one is someone like the suspect. (For example: a DNA test might indicate with z% probability that the guilty one is someone with blood like the suspect has.)

This means that with z% probability (in this case, 95%) we have recognized what the guilty one is like. Who the guilty one actually is is a different question and has different probabilities.
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Old 25th April 2011, 01:46 AM   #66
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JJM,

I didn't think it was possible for you make less sense than you previous post.
You proved me wrong.

Please forget about your solution for a moment
Think about how the correct solution is derived:

Here is the 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?

In other words, what is the probability that the positive result obtained by the machine is actually a true positive?

Here is the solution:

P = P
true positive/(Ptrue postiive + Pfalse positive)
P
true postiive = 1/100 x 95/100
P
false positive = 99/100 x 5/100
P = .161016949 or 16.1016949%


Can it be any simpler?
And as soon as you understand that...bonus: you immediately realise why your solution is wrong.


Last edited by BillyJoe; 25th April 2011 at 01:49 AM.
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Old 25th April 2011, 01:55 AM   #67
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Let's ignore false negatives and use a slightly different method.

You're a doctor in the 80'ies, you just discovered a useful way to detect AIDS. Problem is, 5% of the times it gives a false positive. It will never, however, give a false negative. In the population you're testing, you know that 1 in 1000 people have AIDS.

If you get a positive result, what is the chance that the tested person has AIDS?

Well, it clearly can't be 95% - the test is 100% reliable when testing an infected person. So we must take into account all people that would come out as positives over 1000 trials, 1000 being the average amount of trials we would have to run to get a person with AIDS.

Chance for true positive given any test = 1/1000 or 0.001
Chance for false positive given any test = 5/100 or 0.05

Thus
P = 0.001/(0.001 + 0.05) =~ 0.02

So in reality, given a positive, the odds that it's a true positive is just below 2%.

I hope this helps.
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Old 25th April 2011, 03:35 AM   #68
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Originally Posted by TubbaBlubba View Post
Chance for false positive given any test = 5/100 or 0.05
That should be (5/100)(999/1000).

(But this has only a very small effect on the final answer, changing it from 20/1020 to 20/1019, i.e., from about 1.9608% to about 1.9627%.)
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Old 25th April 2011, 03:54 AM   #69
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Originally Posted by JJM 777 View Post
Others think that the 95% probability of the test result being correct somehow _changes_ when we think about how many different people could possibly be included within the 5% of false positives. This viewpoint does not hold water, methinks: the probability of the test being correct is given as 95%, so when you have as many suspects as you ever wish, and you do the test 100 times, 10,000 times or 1000,000 times, you still get the correct indication with 95% of takes and incorrect indication with 5% of takes.
Yes, you get a correct indication 95% of the time you administer the test. However, in general, you do not get a correct indication 95% of the time that the test says "lying", nor do you get a correct indication 95% of the time that the test says "telling the truth". The 95% figure is just for all tests combined.

Originally Posted by JJM 777 View Post
Thus the probability of the indication being correct is 95%.
No, because in this problem, we already know that the test said "lying", so to get the right answer, we need to focus only on tests that say "lying", not on all tests regardless of result. In what fraction of tests that say "lying" is the subject lying? That's the answer being sought.
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Old 25th April 2011, 06:31 AM   #70
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Originally Posted by TubbaBlubba View Post
Let's ignore false negatives and use a slightly different method.
What difference does it make?
False negatives don't even come into the calculation

Quote:
Chance for true positive given any test = 1/1000 = 0.001
Ptrue positive = 1/1000 x 95/100 = 0.00095

Quote:
Chance for false positive given any test = 5/100 = 0.05
Pfalse positive = 999/1000 x 5/100 = 0.04995

Quote:
Thus
P = 0.001/(0.001 + 0.05) =~ 0.02

P = 0.00095
/(0.00095 + .04995) = 0.0186640472

Quote:
So in reality, given a positive, the odds that it's a true positive is just below 2%.
Actually 1.8664 %

Quote:
I hope this helps.
Well, I hope it does.
In this case, of course, it didn't make much difference, but that won;t always be the case.

Last edited by BillyJoe; 25th April 2011 at 06:34 AM.
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Old 25th April 2011, 08:25 AM   #71
69dodge
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Originally Posted by BillyJoe View Post
Ptrue positive = 1/1000 x 95/100 = 0.00095
How'd you get that?

In the example, one in a thousand people have AIDS, and for them the test will always come back positive. So it should be simply 1/1000.
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Old 25th April 2011, 08:26 AM   #72
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The problem as stated in the OP makes no mention of whether the machine's errors are equally likely to be false positives or false negatives. Wouldn't that make a big difference?
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Old 25th April 2011, 08:29 AM   #73
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Originally Posted by TubbaBlubba View Post
If you get a positive result, what is the chance that the tested person has AIDS?

Well, it clearly can't be 95% - the test is 100% reliable when testing an infected person.
Not sure what you mean here.

If 19 out of every 39 people had AIDS, the answer would be 95%.
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Old 25th April 2011, 08:59 AM   #74
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Originally Posted by Prometheus View Post
The problem as stated in the OP makes no mention of whether the machine's errors are equally likely to be false positives or false negatives. Wouldn't that make a big difference?
Yes, it would make a difference.

It seems reasonable to interpret a statement like "the lie detector correctly classifies 95% of people" as meaning that the lie detector correctly classifies 95% of liers and also 95% of truth-tellers, because otherwise the statement would need to include an assumed distribution of liers vs. truth-tellers in order to give any figure at all. The more liers there are, the closer the overall figure would be to the figure for liers alone and the farther it would be from the figure for truth-tellers alone, and vice versa. Only if the two figures are identical does it not matter how many liers there are vs. truth-tellers.
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Old 25th April 2011, 04:29 PM   #75
BillyJoe
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Originally Posted by 69dodge View Post
How'd you get that?

In the example, one in a thousand people have AIDS, and for them the test will always come back positive. So it should be simply 1/1000.

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Old 25th April 2011, 10:05 PM   #76
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Originally Posted by 69dodge View Post
Not sure what you mean here.

If 19 out of every 39 people had AIDS, the answer would be 95%.
95.49% to be precise but this is a situation where the number of people with aids is very nearly equal to the number of people without aids. When the proportion of people with aids decreases, so does the probability that a positive reading is a true positive.
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Old 25th April 2011, 10:15 PM   #77
69dodge
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Originally Posted by psionl0 View Post
95.49% to be precise
How'd you get that? I think it's exactly 95%.

I solved
$$\frac{p}{p + \frac{1}{20}(1 - p)}} = \frac{19}{20}$$
for p, and got 19/39.

Originally Posted by psionl0 View Post
but this is a situation where the number of people with aids is very nearly equal to the number of people without aids. When the proportion of people with aids decreases, so does the probability that a positive reading is a true positive.
Yes, I agree.
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Old 25th April 2011, 10:18 PM   #78
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Here is a thought. They test the 99 innocent people. About 5 of them will test positive, as well as the 95% of the guilty party. So you end up with about 6 people with positive results. Now you work the rest out.
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Old 25th April 2011, 11:07 PM   #79
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Originally Posted by 69dodge View Post
How'd you get that? I think it's exactly 95%.
Sorry, a typo.
I originally entered
(19/35 *1) / (19/39*1 + 20/39*0.05) = 0.95487

when I should have entered
(19/39*1)/(19/39*1+20/39*0.05) = 0.95.
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Old 25th April 2011, 11:19 PM   #80
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Originally Posted by rjh01 View Post
Here is a thought. They test the 99 innocent people. About 5 of them will test positive, as well as the 95% of the guilty party. So you end up with about 6 people with positive results. Now you work the rest out.
That's the best layman explanation I have seen on this thread.

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