psionl0
Skeptical about skeptics
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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Very hard question in online test: all answers are wrong?
- Thread starter Ivor the Engineer
- Start date
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.
Yes.
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.)
EeneyMinnieMoe
Philosopher
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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.
)
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.
)Roger Ramjets
Philosopher
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.
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.
Roger Ramjets
Philosopher
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!
Aepervius
Non credunt, semper verificare
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.
The machine will identify approximately 6 criminals but approximately 5 of these identifications will be wrong.
psionl0
Skeptical about skeptics
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.
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blobru
Philosopher
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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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rjh01
Gentleman of leisure
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.
TubbaBlubba
Knave of the Dudes
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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.
I got all "understanding risks" questions right except I miscalculated the odds of the dice coming up 6, apparently.
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%
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%
TubbaBlubba
Knave of the Dudes
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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%."
Professor Yaffle
Butterbeans and Breadcrumbs
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!
Yes; Social was my biggest risk area, and that was Low.
Same here; maybe the phrasing threw me.
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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Professor Yaffle
Butterbeans and Breadcrumbs
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.
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.
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Subduction Zone
Muse
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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.
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.
Ziggurat
Penultimate Amazing
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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.