witness answer

[cha]

If 80% of the population wears red, and the witness can only correctly identify 85% of them, 85% of the original 80% equals 68%.
If he can only correctly idenify 85% of the 20% who wear green, then only 17% of the civilians can be identified.
Add these for a total of 85%. The witness can only be 85% accurate, not enough to convict.
But this was given in the problem. Where is the puzzle?

[JediSoop]

Here's my take on it...I'm not sure if the logic is sound so tell me if you see anything fishy:

If 20% of the population is civilian, then that means there is (obviously) a twenty percent chance that the guilty person was civilian.

Now, there is a 15% chance that the witness was wrong in his identification.

1. If you compound these possibilities, the possibility that the guilty person was civilian AND that the witness was wrong is (15% * 20%) 3%

2. That makes (20% * 85%)=17% chance that the guilty was civilian and the witness was right (this case did not happen however)

3. (80% * 85%)=68% chance that the guilty was a soldier and the witness was right in indentification

4. (80% * 15%)=12% chance that the guilty was a soldier and the witness was wrong (this was also not the case).

Now, combine all the possibilities that are beneficial for the prosecution (cases 2,3, and 4) and you get 97%...greater than the 95% needed.

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One ring to rule them all...



[This message has been edited by JediSoop (edited 08-29-2000).]

[JediSoop]

Or, to put it simpler, Case 1 was the ONLY case which the defense was trying to prove...and because it was only 3% possible, anything else would be 97% possible, good enough for the prosecution

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One ring to rule them all...

[mithrandir]

ok, first if you are going to use this, you have to throw out the two cases that you know didn't happen. so, all that's left is 68% and 3%. so now we have 68/71=95.8%, still above 95%.

i'm still not sure if you can really do that in probability, it's been a while now, i'm getting rusty, but i'm sure that's what the prosecution used.

[Ghost Post]

Mithrandir is correct.

It's classical conditional probability.

Bayes rule P(A|B) = P(A) * P(B|A) / P(B)

In English, the probability of A given B is equal to the probability of A times the probability of B given A, divided by the probability of B

Here A is "the criminal wore red", and P(A) is 0.80

and B is "the witness identifies the criminal as having worn red", and P(B) is 0.71 (0.85*0.8 + 0.15*0.2)

So P(A|B) is the probability that the criminal wore red given the identification by the witness, and it is 0.8 * 0.85 / 0.71 = 0.9577

Of course, I think to be fair, the following defense argument would have to be taken seriously:

If a soldier committed the crime, we hold the commanding General responsible, because we expect him to have a significant and positive controlling influence over his soldiers. So the question is, did he in fact exert such an influence, and was it adequate? We can't presume, from the start, that the Commanders positive influence over the men was inadequate, and we certainly can't presume, from the start, that the Commander had no such influence over his men whatsoever. That would be a presumption of guilt from the start.

The probability, all witnessing testimony aside, that the criminal wore red can't be assumed to be 80%, as it has been in the prosecutions argument. 80% is the probability of a randomly selected individual wearing red. To assume that the probability the criminal wore red is 80% is to assume that a soldier and a civilian are equally likely to commit murder. And to assume that is to assume the commanding General has no positive controlling influence over his soldiers whatsoever. If we assume that, then we are presuming the General guilty from the start.

Therefore, the prosecutions argument essentially concludes guilt by first assuming it. It is a circular argument, totally without merit.


[This message has been edited by extro... (edited 08-29-2000).]

[daniel801]

I can't do the fill-in-the-blank math part (X), but it's computable. Here's what the defense needs to say:

First, explain how it's 95.77% confidence level, then explain that there are only X citizens (a small number) and due to rounding, the confidence level drops down to Y people and thus one is not 95% confident.

Just to be clearer, try doing this with 5 citizens and work your way up.

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anger is a gift

[HyToFry]

Guilty! HA! Thats the answer.. not 95.8 or 98 or 97..

It's "GUILTY"

ppllibbbbbb =þ

[giarc]

The witnesses testimony must prevail.
We know that the deceased was a civilian so he must have been wearing green - as the puzzle says ALL civilians wear green.
It is only necessary to ask the witness if the assailant was wearing the same colour or different to the victim.
Same Colour - Innocent
Different colour - Guilty

[JediSoop]

mithrandir- yeah thanks That sounds right to me. I knew there was somthing hazy in my reasoning. Props to HyToFry and giarc for some pretty cool answers too...hehe

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One ring to rule them all...

[mole]

The prosecution actually had 3 people there, then, when the witness chose, one was taken away, and the witness was offered another choice.

This will only give 67% accuracy though, so the prosecution then went to find 17 more people. Since there are only 5 people in the puzzle (pointed out by dave801), the witness can only ever be 80% sure.

[daniel801]

maybe mole thinks dave10000 and I are a composite; dave's pretty smart and so if we were one person we would be one good-looking einstein.

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anger is a gift

[Ghost Post]

The way I see it is this. Either the witness had his eyesight working correctly at the time of the murder or he didn't.

The chance that it was working correctly is 85%.

So the chance he might be mistaken is 15%. However, 80% of his country men were red wearing soldiers. So that the chance that his eyes were playing up but that it was still a red wearing soldier is 80/100 * 15 = 12.

So the odds of it being a red wearing soldier is 97%.

Note : Keep in mind that for 15% of the time the witness can't distinguish red from green. That isn't to say 15% of the time he mistakes red for green or vice versa.

Oh, and giarc, the puzzle states that the Puzzleanian's were wearing red/green. Not the unnamed enemy.

[mole]

I only had an 85% chance of telling you two apart.... and when I couldn't I didn't want to guess...

[hey_herb]

Here is another way to look at it.

The witness sees 71% of the population in red.
85% of 80 = 68 wearing red and seen as red.
15% of 20 = 3 wearing green but seen as red
total = 71 seeing red

Therefore, if the witness says he saw red, he has a 68 of 71 chance of being right or the 95.77% discussed earlier.

He would actually be less credible if he said he saw green. His chance of being right then drops to 58.6%.

15% of 80 = 12 wearing red but seen as green.
85% of 20 = 17 wearing green and seen as green.
total = 29 seen as green.

So if the witness said he saw green he would only have a 17 of 29 chance of being right.

Too bad 80% of the population were soldiers. A few less and the witness wouldn't have been reliable enough.

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Smiles are infectious.

[ZenBeam]

Extro's analysis is correct: the prosecution hasn't provided evidence that soldiers are equally likely to have killed the victim as civilians. Just assuming

[Ghost Post]

Thanks, ZenBeam. And a clarification: That sort of reasoning CAN be valid, and is used in courts all the time. For instance, with blood types. It's quite reasonable to assume a person with type A blood is equally likely to murder as a person with type O. There is just no reason to believe blood type has any bearing on ones likelihood of commiting murder. But in this puzzle, the General would be held accountable for his men, and their is reason to believe he would attempt to control them so that they would be less likely to commit murder than the average citizen. And again, to assume they are no less likely is to assume the General had no control or positive innfluence over them.

[Nil]

Can the prosecution just ask the witness what color the victim was wearing? As long as the killer was of a different color as the victim at that time (which disregards color blindness) we know that the killer is different from the victim. Since victim is stated as a civilian, thus won't killer have to be a soldier?

[Gill Bates]

I tend to believe the P(A|B) solution.

Someone mentioned that the color of the soldier could be determined by asking the witness if he was wearing the same color as the victim. But color blindness causes colors to look the same, not to look wrong or different.

Here is an interesting article someone sent me on color-blindness. It's slanted towards designing color safe web pages, but it does a nice job of explaining the problem. http://www.webtechniques.com/archives/2000/08/newman/

[munira]

Just for clarification:

P(A/B)= P(AUB)/P(B)

[Ghost Post]

Yeah, I geuss it would be. P(AUB) would have to equal P(A) * P(B|A) (but not necessarily equal P(A)*P(B), as in when A and B are not independent).

I always check my formulas at http://mathworld.wolfram.com/

Great site!

[Ghost Post]

Actually, that shouldn't be AUB but A&B ('U' being union, or "or", and '&' being intersection, or "and" - an upside down U is usually used)

[Da5id]

The key question is how many trials were run to determine his accuracy of 85%? Anyone remember the formula to determine the possible error of a sample? I know it's not SD it's more complex. If the accuracy could be as low as 82.6% then he would be not guilty.

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Wake up, it's time to die.

[This message has been edited by Da5id (edited 09-02-2000).]

[Ghost Post]

This is an example of "Base-rate Neglect". The most famous example is given by Stephen Stitch in "Is MAn a Rational Animal" (1998, recently in Philosophical Bridges, Kolak, ed.). This question was given to a group at Harvard Medical School:

"If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person's symtoms or signs?"

Most of the group responded that the chance was 95% -- making the same mistake that the general makes... the base rate must be considered in both cases. The correct answer is 2%.

Likewise, the base rate must be considered in the general's case. For the 15% of the time that the witness is unreliable, there is an 80% chance that the assailant was a soldier. So there is a 12% chance that the witness was unreliable and the assailant was a soldier. For the 85% of the time when the witness is reliable, there is a 100% chance that the assailant is a soldier. So there is an 85% chance that the witness was reliable and the assailant was a soldier. Since all that is important is the probability that the assailant was a soldier, we can add these two probabilities, giving a 97% chance, enough to convict.

[Fudgesicles on Uranus]

How do they know that its not a different soldier? Hey it could happen.

[Quailman]

Philonius: You can't add the probabilities in the manner you describe. There are four outcomes of the witness' identification.

Red ID'ed correctly as Red
Red ID'ed incorrectly as Green
Green ID'ed correctly as Green
Green ID'ed incorrectly as Red

The probabilities of these 4 outcomes are:

.8 x .85 = .68
.8 x .15 = .12
.2 x .85 = .17
.2 x .15 = .03

Note that these sum to 1.00. As you can see, 71% (.68+.03) of the time he will identify a random individual as wearing Red, which he in fact did. Of these instances, (68/71), or .95775 of the time he will have been correct.

[cha]

I now agree with the 97% answer and withdraw my original comment. This puzzle raises another interesting real life question. In an earlier response, it was stated that if a test for a disease produces a 5% false positive, and 1/1000 people actually have the disease, then only 2% of the positive responses will be accurate. Applying this to pre-employment drug tests, if we assume that a 1% false positive rate is acceptable, and that one in a hundred people applying for a particular job actually have detectable drugs in their system, then out of 10,000 tests, 100 would accurately test positive, and 99 more would test falsely positive. So nearly 50% of the positive results would be false. The "common sense" interpretation of the odds could have very unfair results for a great many people.

[Ghost Post]

Uh, Cha, the 97% figure is also incorrect. Several people have correctly given the reason why the actual answer is 95.77%.

[theaust]

What evidence have you got that the propensiity to kill is even distributed among the population ie the basis of your assumption that there is a 20% chance a civilian was involve?
The only evidence we have is:
that there is an 85% chance that the witness's statement was reliable; and
the witness claims the color was red.
We know nothing about the distribution of murderers among the reds and greens.

[Ghost Post]

Yeah, my point exactly.

If you assume 80% of muderers will be soldiers, and that that is beyond the control of the commmanding General, how can you hold him accountable?

And if you assume it is within his power to influence the likelihood of a soldier committing murder, then you can't assume that 80% of murderers are soldiers. That would assume the General exerted no influence, and you may as well declare him guilty without ever looking at any numbers.

[dave10000]

ZenBeam's reasoning (I know his PIN number, by the way ) a few posts above is exactly right. Without knowing the base rate of how likely it is that a civilian as opposed to a soldier would kill in the particular instance, the 80/20 base rate is only a small part of the story. What if the killing took place in Civilianville, where half the immediate residents are civilians? Or if the victim was someone who was especially hated by civilians. Or even let's look at the total number of renegade killers -- if statistics show that renegades have committed more than half the murders in the past 10 years, surely that would be relevant evidence to take into account.

[WARNING -- REAL LIFE DISCUSSION FOLLOWS] In the law, the evidence of the sort given in the problem is often referred to as "naked statistical evidence." And many courts refuse to allow a finding of criminal or civil liability based solely on naked statistical evidence -- largely for policy reasons (such as: even if there is a 95% chance this guy did it, we want evidence actually tying this guy to this crime, rather than letting him possibly be the victim of statistics. It's better to let him go than to convict him on "bad luck" or on evidence the public might think is "unfair").

A similar situation was discussed in the case of Smith v. Rapid Transit, Inc., 1945, which raised the "Blue Bus Hypothetical" that has been much discussed in legal circles ever since. Roughly: If Blue Bus Company runs 80% of the busses in the area, and Witness testifies she was sun down by a blue bus, should Blue Bus company be held liable? Sure, there may be an 80% chance they did it, but that means they'll be liable in 100% of the cases. And isn't also a little fishy that there is NO other evidence that the plaintiff provided to indicate that it was BBC's fault. Statistics IN CONJUNCTION WITH other evidence can be persuasive, but pure statistics alone -- courts don't like making rulings that way, even if LOGICALLY such rulings make sense.

I could go on, but I won't. There's a LOT of information on how the law has handled these types of situations, and most of it is very interesting. Here's one article, available on the internet, that discusses the issue to some degree:
http://www.monash.edu.au/journals/psycoloquy/volume_5/psyc.94.5.33.base-rate.14.wells-windschitl

If you are still interested, look up Summers v. Tice and Sindell v. Abbott Laboratories, both of which are slightly off the point of the present question, but interesting and informative.

Bottom line solution to the Puzzle as given: On the evidence as presented, if no other relevant evidence existed, there is a 96% chance that the killer was a soldier, for reasons discussed above. However, such evidence constitutes "naked statistical evidence," and there certainly are many other base rates not before the court (malpractice by defense attorney?) that are highly relevant and could substantially alter the ultimate probability -- thus, no REAL court would convict on the evidence presented. But of course, I can't say what they'd do in Puzzleania.

[hjltax]

Here's a different take on the puzzle. The General & his defense team should have argued a different point of law in this case.

Here is the first paragraph of the puzzle:

At the end of the last century, a General of Puzzlania's army was court martialed. The accusation was brought forth that a Puzzlanian soldier had killed an unarmed civilian in an opposing army. The General was considered responsible for the conduct of all his commands.

The defense should have agued that the killed man could not have been a civilian if he was indeed "in an opposing army." Assuming that it is not against the law to kill a member in an opposing army (armed or unarmed), it should have been argued that no law was in fact broken.

[Gigantor]

Isn't the fact that he can differentiate between green and red only 85% of the time irrelevant? We already know that a civilian was killed, and since he managed to report that a soldier had done the killing, then he managed to differentiate the two uniforms appart. If he had failed he would have reported that a civilian had killed the civilian.

[Ghost Post]

Quailman:

The question says, "the witness could correctly tell red and green apart with only 85% accuracy."

I take this to mean that the other 15% of the time the witness is unreliable, not WRONG. It is ambiguous, I suppose, but it seems to me that in those 15% of times when the witness is unreliable the witness is actually colorblind -- if this is the case then my answer is correct. If in those 15% of the times when the witness is inaccurate she ALWAYS gives the wrong answer, then your answer is correct. Check it...


[This message has been edited by philonius (edited 09-12-2000).]

[Quailman]

philonius: That's how I saw it. I'm not colorblind, but sometimes I do wear blinders. I took it to mean that the witness always sees a color, but 15% of the time he sees the wrong color.

[Beartalon]

I understand the math, that gives up to 95.8%. All the possibilities are available, some less probable than others, given ONLY that the witness can distinguish red from green 85% of the time. However, the puzzle also states that he can tell the difference between a FAKE uniform and a real one, and there were no confusing light problems. That implies he knows what the counterfeits are, then he knows what else besides color makes a uniform REAL.
In that case:
1. If the killer was a vigilante, he'd have to be in an exact replica of the uniform except for the colour for the witness to identify him as being part of the army. He'd also have to have the same weapons and other paraphernalia. If he wore an army uniform colored green, he would have been very obvious to the rest of the army and civilians that he didn't belong, so there would have been MORE witnesses. So it makes sense that the killer must have been wearing red, in a proper uniform and indistinguishable from a soldier to EVERYONE, including the witness.

That's gives me 100% for wearing red and being a soldier (fake or not), but please show me any holes in my argument. - BearTalon

[Kripa]

I believe the 95.77 answer is correct. Just wanted to write down the problem in conventional probability theory notation.

Define 2 Discrete Random Variables.

AnyManRed Defined as 1 if man is red and 0 otherwise.
P(AnyManRed==1)=0.8
P(AnyManRed==0)=0.2

WitnessSeesRed Defined as 1 if the witness saw red and 0 otherwise.
P(WitnessSeesRed==1)=0.8*0.85+0.2*0.15=0.71
P(WitnessSeesRed==0)=0.8*0.15+0.2*0.85=0.39

We are interested in the
I abondon the ==1 stuff since they are binary R.Vs
P(AnyManRed | WitnessSeesRed) = P(AnyManRed,WitnessSeesRed)/P(WitnessSeesRed)

P(AnyManRed,WitnessSeesRed) can be read as the Man being red and witness seeing red. this is clearly 0.8*0.85 = 0.68 Note that the events are dependant and hence not P(AnyManRed)*P(WitnessSeesRed)

hence P(AnyManRed | WitnessSeesRed)=0.68/0.71=95.776%

or the probability that the man who the witness saw as red was indeed red is greater than that required to hang the general.

[JF]

Hey, I'm just a visitor, but there's lots of reasons to acquit the poor general.

1) As stated in the puzzle, the murder victim was a civilian in ANOTHER culture. No evidence is presented as to what color shirts people in THAT culture wear. It's entirely possible that in Conundrumia, all convicted murderers wear red shirts once paroled. OK, it doesn't have to be that extreme, but you get the point. The redshirt/blueshirt evidence only applies to Puzzlania. The murder was not in Puzzlania. Acquittal.

2) As has already been mentioned, the victim was apparently in an opposing army. Even if he was a civilian, his presence in the army made him a reasonable target in a wartime situation. No crime has been committed. Acquittal.

3) All of this discussion about the "reliability" of the witness' eyesight depends on a big assumption about the meaning of "accuracy". There are at least 2 relevant--and contrary--definitions of accuracy here: sensitivity and specificity. The Steven Stich example above appeals only to specificity, as does the puzzle itself. But maybe the witness' "accuracy" is actually sensitivity, and not specificity?

To explain: Specificity measures the rate of false POSITIVES, while sensitivity measures the rate of false NEGATIVES.

Without information about what this "85%" figure means, you've gotta acquit. And the prosecutor, who has failed to elicit this important information during direct examination, gets reprimanded and sent back to litigator's school

John

[Ghost Post]

It's strange that people have so much trouble with probability problems. The solution given for this problem is wrong,
but nobody has given the correct answer.

"Philonius" wrote:

> This is an example of "Base-rate Neglect".
> The most famous example is given by
> Stephen Stitch in "Is Man a Rational
> Animal" (1998, recently in Philosophical
> Bridges, Kolak, ed.). This question was
> given to a group at Harvard Medical School:
> "If a test to detect a disease whose
> prevalence is 1/1000 has a false positive
> rate of 5%, what is the chance that a
> person found to have a positive result
> actually has the disease, assuming you
> know nothing about the person's symtoms or
> signs?"

> Most of the group responded that the
> chance was 95% -- making the same mistake
> that the general makes... the base rate
> must be considered in both cases. The
> correct answer is 2%.

I'm familiar with this problem.

> Likewise, the base rate must be considered
> in the general's case.

No. The base rate is irrelevant here. We've got an actual witness with a measured rate of accuracy.

> For the 15% of the time that the witness
> is unreliable, there is an 80% chance that
> the assailant was a soldier. So there is a
> 12% chance that the witness was unreliable
> and the assailant was a soldier.

No. The witness is accurate 85% of the time. In measuring his accuracy we cannot read his mind and determine which of his correct answers were arrived at by chance. In the whole other 15% of cases, the witness gets the color wrong. This is what a percentage of accuracy means.

The remainder of Philonius' analysis is similarly incorrect (though, aside from the incorrect assumptions, it's well-reasoned and well-expressed).

Bear Talon is right that the problem seems to assume that the only difference between soldiers and civilians is the color of their clothes, clearly an unrealistic assumption.

And JF is right about the assumption about the colors of the clothing of the enemy.

But, even leaving these objections aside, it has been assumed that the witness correctly identified the victim as a civilian. But he could as easily be wrong about that, too. The general is guilty only if the perpetrator was a soldier AND the victim was a civilian. The probability of the witness being correct about each of these things separately is 85%, so the overall probability that the general is guilty is 85% x 85% = 72.25% and if he was found guilty he has been the victim of a grievous miscarriage of justice. Even if it is assumed that the victim was a civilian, the probability is still only 85% and the general should still be acquitted.

Kevin Langdon
kevin.langdon@polymath-systems.com http://www.polymath-systems.com

[Vanyo]

[Ghost Post]

I believe the 95.77 answer is correct. Just wanted to write down the problem in conventional probability theory notation.
Define 2 Discrete Random Variables.

AnyManRed Defined as 1 if man is red and 0 otherwise.
P(AnyManRed==1)=0.8
P(AnyManRed==0)=0.2

WitnessSeesRed Defined as 1 if the witness saw red and 0 otherwise.
P(WitnessSeesRed==1)=0.8*0.85+0.2*0.15=0.71
P(WitnessSeesRed==0)=0.8*0.15+0.2*0.85=0.39

We are interested in the
I abondon the ==1 stuff since they are binary R.Vs
P(AnyManRed | WitnessSeesRed) = P(AnyManRed,WitnessSeesRed)/P(WitnessSeesRed)

P(AnyManRed,WitnessSeesRed) can be read as the Man being red and witness seeing red. this is clearly 0.8*0.85 = 0.68 Note that the events are dependant and hence not P(AnyManRed)*P(WitnessSeesRed)

hence P(AnyManRed | WitnessSeesRed)=0.68/0.71=95.776%

or the probability that the man who the witness saw as red was indeed red is greater than that required to hang the general.


ta da,thank u very much