|
Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today. |
24th April 2011, 11:01 AM | #41 |
Knave of the Dudes
Join Date: Jul 2010
Posts: 12,936
|
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. |
__________________
"The president’s voracious sexual appetite is the elephant that the president rides around on each and every day while pretending that it doesn’t exist." - Bill O'Reilly et al., Killing Kennedy |
|
24th April 2011, 11:06 AM | #42 |
Penultimate Amazing
Join Date: Jun 2003
Posts: 56,422
|
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).
|
__________________
"As long as it is admitted that the law may be diverted from its true purpose -- that it may violate property instead of protecting it -- then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and all-absorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious." - Bastiat, The Law |
|
24th April 2011, 11:13 AM | #43 |
Knave of the Dudes
Join Date: Jul 2010
Posts: 12,936
|
|
__________________
"The president’s voracious sexual appetite is the elephant that the president rides around on each and every day while pretending that it doesn’t exist." - Bill O'Reilly et al., Killing Kennedy |
|
24th April 2011, 01:05 PM | #44 |
I would save the receptionist.
Moderator Join Date: Jul 2006
Location: Florida
Posts: 28,352
|
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. |
__________________
I have the honor to be Your Obdt. St L. Leader |
|
24th April 2011, 01:12 PM | #45 |
Knave of the Dudes
Join Date: Jul 2010
Posts: 12,936
|
"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. |
__________________
"The president’s voracious sexual appetite is the elephant that the president rides around on each and every day while pretending that it doesn’t exist." - Bill O'Reilly et al., Killing Kennedy |
|
24th April 2011, 03:23 PM | #46 |
Penultimate Amazing
Join Date: Feb 2006
Posts: 29,167
|
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. |
24th April 2011, 03:32 PM | #47 |
Illuminator
Join Date: Feb 2011
Posts: 4,302
|
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. |
24th April 2011, 03:51 PM | #48 |
Philosopher
Join Date: Nov 2004
Posts: 7,508
|
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%. |
__________________
I don't need to fight to prove I'm right. - Baba O'Riley |
|
24th April 2011, 04:05 PM | #49 |
Knave of the Dudes
Join Date: Jul 2010
Posts: 12,936
|
|
__________________
"The president’s voracious sexual appetite is the elephant that the president rides around on each and every day while pretending that it doesn’t exist." - Bill O'Reilly et al., Killing Kennedy |
|
24th April 2011, 04:33 PM | #50 |
Penultimate Amazing
Join Date: Jun 2003
Posts: 56,422
|
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.
|
__________________
"As long as it is admitted that the law may be diverted from its true purpose -- that it may violate property instead of protecting it -- then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and all-absorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious." - Bastiat, The Law |
|
24th April 2011, 05:04 PM | #51 |
Illuminator
Join Date: Jul 2008
Posts: 4,852
|
|
__________________
It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ |
|
24th April 2011, 05:21 PM | #52 |
Thinker
Join Date: Jul 2008
Posts: 174
|
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. |
24th April 2011, 05:33 PM | #53 |
Illuminator
Join Date: Jul 2008
Posts: 4,852
|
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.
|
__________________
It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ |
|
24th April 2011, 05:48 PM | #54 |
Philosopher
Join Date: Jul 2005
Posts: 6,372
|
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:
Quote:
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.) |
24th April 2011, 06:02 PM | #55 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
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. |
24th April 2011, 06:26 PM | #56 |
Philosopher
Join Date: Jun 2008
Posts: 7,110
|
|
__________________
We don't want good, sound arguments. We want arguments that sound good. |
|
24th April 2011, 08:34 PM | #57 |
Master Poster
Join Date: Jul 2007
Posts: 2,456
|
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. |
24th April 2011, 09:56 PM | #58 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
|
24th April 2011, 11:04 PM | #59 |
Penultimate Amazing
Join Date: Aug 2001
Posts: 12,531
|
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? |
24th April 2011, 11:07 PM | #60 |
Penultimate Amazing
Join Date: Aug 2001
Posts: 12,531
|
...oops, I didn't see ThunderChucky's and 69Dodge's responses
|
24th April 2011, 11:20 PM | #61 |
Suspended
Join Date: Jun 2004
Posts: 4,060
|
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%. |
24th April 2011, 11:31 PM | #62 |
Illuminator
Join Date: Jul 2008
Posts: 4,852
|
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!
|
__________________
It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ |
|
24th April 2011, 11:48 PM | #63 |
Penultimate Amazing
Join Date: Feb 2005
Location: Shanghai
Posts: 16,040
|
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%. |
__________________
"... when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together." Isaac Asimov |
|
24th April 2011, 11:55 PM | #64 |
Philosopher
Join Date: Sep 2006
Posts: 6,985
|
|
25th April 2011, 12:04 AM | #65 |
Suspended
Join Date: Jun 2004
Posts: 4,060
|
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. |
25th April 2011, 01:46 AM | #66 |
Penultimate Amazing
Join Date: Aug 2001
Posts: 12,531
|
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 = 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% Can it be any simpler? And as soon as you understand that...bonus: you immediately realise why your solution is wrong. |
25th April 2011, 01:55 AM | #67 |
Knave of the Dudes
Join Date: Jul 2010
Posts: 12,936
|
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. |
__________________
"The president’s voracious sexual appetite is the elephant that the president rides around on each and every day while pretending that it doesn’t exist." - Bill O'Reilly et al., Killing Kennedy |
|
25th April 2011, 03:35 AM | #68 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
|
25th April 2011, 03:54 AM | #69 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
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.
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. |
25th April 2011, 06:31 AM | #70 |
Penultimate Amazing
Join Date: Aug 2001
Posts: 12,531
|
What difference does it make?
False negatives don't even come into the calculation
Quote:
Quote:
Quote:
P = 0.00095/(0.00095 + .04995) = 0.0186640472
Quote:
Quote:
In this case, of course, it didn't make much difference, but that won;t always be the case. |
25th April 2011, 08:25 AM | #71 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
|
25th April 2011, 08:26 AM | #72 |
Acolyte of Víðarr
Join Date: Nov 2007
Posts: 50,572
|
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?
|
__________________
As Einstein once said, "If you can't think of something relevant to say, just make something up and attribute it to some really smart dead guy." "I find your lack of pith disturbing," - Darth Rotor .......... Don't be offended. I'm not calling you a serial killer. -- Ron Tomkins. |
|
25th April 2011, 08:29 AM | #73 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
|
25th April 2011, 08:59 AM | #74 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
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. |
25th April 2011, 04:29 PM | #75 |
Penultimate Amazing
Join Date: Aug 2001
Posts: 12,531
|
|
25th April 2011, 10:05 PM | #76 |
Skeptical about skeptics
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 20,952
|
|
25th April 2011, 10:15 PM | #77 |
Illuminator
Join Date: Nov 2002
Posts: 3,607
|
|
25th April 2011, 10:18 PM | #78 |
Gentleman of leisure
Tagger
Join Date: May 2005
Location: Flying around in the sky
Posts: 28,092
|
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.
|
25th April 2011, 11:07 PM | #79 |
Skeptical about skeptics
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 20,952
|
|
25th April 2011, 11:19 PM | #80 |
Skeptical about skeptics
Join Date: Sep 2010
Location: 31°57'S 115°57'E
Posts: 20,952
|
|
Thread Tools | |
|
|