Very hard question in online test: all answers are wrong?

[TubbaBlubba]

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.

 

[Ziggurat]

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).

 

[TubbaBlubba]

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.

 

[Loss Leader]

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.

 

[TubbaBlubba]

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.

 

[marplots]

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.

 

[ehcks]

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.

 

[DevilsAdvocate]

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%.

 

[TubbaBlubba]

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.

 

[Ziggurat]

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.

 

[Perpetual Student]

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.

 

[Illustronic]

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.

 

[Perpetual Student]

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.

 

[Rasmus]

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.)

They make it easy by having 100 people, the question is bogus.

How is the question bogus?

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.)

 

[69dodge]

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.

 

[Roger Ramjets]

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!

 

[ThunderChunky]

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.

 

[69dodge]

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.

 

[BillyJoe]

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 = P_{true positive}/(P_{true postiive} + P_{false positive})
P_{true postiive} = 1/100 x 95/100
P_{false 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?

 

[BillyJoe]

...oops, I didn't see ThunderChucky's and 69Dodge's responses