Statistics puzzle - answers by Friday

[callipygian]

I didn't watch the video, but I'll throw in my analysis.

I'm going to assume "99% accurate" means 99% of the counters/diseased (they're one and the same, anyway :laugh: ) are identified, and 99% of the non-counters are cleared.

So if X fraction of the population are counters, then the fraction of the population who tests positive are 0.99*X+0.01*(1-X) = 0.01+0.98*X. The percentage of people who are counters who test positive would be f = 0.99*X/(0.01+0.98*X).

For an X of 0.01, f would equal 0.5, or a 50% chance that the postive is a counter.

For an X of 0.1, f would equal 0.99/1.08, or a 92% chance.

Interestingly enough for the opposite f' (defined as the probability of actually being a non-counter if the test is negative), it's virtually certain.

For an X of 0.01, f' would equal 0.99*0.99/(0.99-0.98*.01) = 0.9999!

For an X of 0.1, f' = 0.99*0.9/(0.99-0.98*.1) = 0.9989!

So it's relatively simple to clear people, but relatively difficult to prove them guilty.

 

[la_dee_daa]

for those of you who don't feel like watching the video.. here is what it goes on to say kinda.


the example it gives is say there is a sample of 1 000 000 people or something and 100 of them have this "disease"

sooo 999900 people dont have it 1% will test positive for it though so thats 9999 people

now our 100 people who have the disease 99 of them will test positive.

so out of the 10098 people who will test positive only 99 really have the disease making the chance.

so like the test is really only 0.98% accurate which ain't good rather then 99% accurate.

we didn't know the % of people who had the disease though which is kinda important

:cow:

 

[vonQuux]

Kind of the point is realising that the question is incomplete..... :joker:

It's worse than that. The question as posed cannot be answered.

For example, the thing about people getting on and off the bus and at the end you're asked how many people are on the bus -- and you're supposed to remember that there is a driver, too -- that's a trick question. But posing a riddle where there is insufficient data to arrive at an answer, or using statements that are ambiguous, that's just a malformed puzzle.

vQ

 

[la_dee_daa]

It's worse than that. The question as posed cannot be answered.

well we can use variables i guess and it would still be true

T= the chance of having the disease
P= accuracy of the test

(P-TP)/(T-2TP+P)= chance of accuracy or what ever

so if we consider that an answear it is answearable but would have helped to know what the chances of getting this disease was.

:cow:

 

[aslan]

If I may rewrite the example to make it a bit more topical (courtesy of Steve Forte):

Consider a software package that is 99% accurate at identifying card counters. Assuming that the casino sees 10,000 blackjack players and 1-out-of-100 of them are actually capable of counting cards, what are the chances of the software correctly identifying a card counter?

-Sonny-

I hope this hasn't been answered in subsequent posts. It's fun to tackle or guess at these posers.

Assuming that those capable of counting were in fact counting, and if by 99% chance you mean that out of a population of 100 counters it will find all but 1, then out of a population of 10,000 it will still find all but 1. This assumes the software has a 100% chance of identifying who the ploppies are.

But if you mean that the software has a 99% chance of determining whether a "player" (could be a counter or a ploppy) is a counter, then the problem is more complex. It means that out of 10,000 there will be 100 mistakes, 99 of them on average wrongly identifying ploppies as counters, and 1 of them wrongly identifying a counter as a ploppy. All told, it will identify 198 as counters (99 accurate, 99 mistaken), and mis-categorize one counter as a ploppy, for a grand total of 100 mistakes out of 1,000, or 1%. All but 1 of the counters would be identified (99%); the only problem is that 99 others would also be labeled as counters who were not. Therefore, you have not identified even 1 counter, because you don't know which of those identified is accurate and which is mistaken. You might say you have a 50/50 chance of guessing which is a counter, but you have no way of actually identifying who is and who isn't.

 

[la_dee_daa]

just imagine all those non counters being kicked out because the casino is 99% accurate on how they identify counters.:joker:

someone show them this and make them stop trying to stop counters knowing that only .98% of the people they kick out are counters assuming the ratios in the video that is. THE MATH PROVES IT:joker:

:cow:

 

[RJT]

It's worse than that. The question as posed cannot be answered.

For example, the thing about people getting on and off the bus and at the end you're asked how many people are on the bus -- and you're supposed to remember that there is a driver, too -- that's a trick question. But posing a riddle where there is insufficient data to arrive at an answer, or using statements that are ambiguous, that's just a malformed puzzle.

vQ

You really have missed the point here vQ. The purpose of the exercise is to show that what seems a relatively simple question statistically, is actually more complex than it look and more of an issue - even with a simple stat like the one in the question - most people, even smart mathematical people, will mis-interprit what conclusions they can draw. But then if you'd watched the lecture i suppose you would know that already....

RJT.

 

[vonQuux]

You really have missed the point here vQ. The purpose of the exercise is to show that what seems a relatively simple question statistically, is actually more complex than it look and more of an issue - even with a simple stat like the one in the question - most people, even smart mathematical people, will mis-interprit what conclusions they can draw. But then if you'd watched the lecture i suppose you would know that already...

Yipes. Looks like I hit a nerve...

Look, asking a question whose implications are deeper than realized at first glance is distinct and entirely different from a question that is ambiguous to the point where five people have five different ideas about what the question actually is asking.

It would be like posing the question about how many people are on the bus after 10 stops without ever mentioning the driver. People start talking among themselves and saying things like:

"Who was driving the bus?"
"Dunno. The puzzle never specifically mentions a driver."
"Maybe it was one of the passengers."
"Well then how did it get to the first stop?"
"So there had to be a driver..."
"Well, if the driver is implied, how do we know they're still on the bus when the count is made??"

Etc, etc.

Someone could show up and say "Aha!, bright people have realized that there is a question about whether or not a driver is part of the count!" Perhaps, but that's not a statistics puzzle, it's an exercise in parsing simple English.

So if you're suggesting your purpose actually was to phrase a question that appeared to ask one thing but whose syntax, when carefully parsed, could be understood as asking several different things, so be it.

But ambiguous syntax is another way of saying "imprecision" and cannot be confused with subtlety.

 

[RJT]

Yipes. Looks like I hit a nerve...

Look, asking a question whose implications are deeper than realized at first glance is distinct and entirely different from a question that is ambiguous to the point where five people have five different ideas about what the question actually is asking.

It would be like posing the question about how many people are on the bus after 10 stops without ever mentioning the driver. People start talking among themselves and saying things like, "Well, who was driving the bus?" "Maybe it was one of the passengers." "Well then how did it get to the first stop?" Etc.

Someone could show up and say "Aha!, bright people have realized that there is a question about whether or not a driver is part of the count!" Perhaps, but that's not a statistics puzzle, it's an exercise in parsing simple English.

So if you're suggesting your purpose actually was to phrase a question that appeared to ask one thing but whose syntax, when carefully parsed, could be understood as asking several different things, so be it.

But ambiguous syntax is another way of saying "imprecision" and cannot be confused with subtlety.

No, you've yet to hit a nerve. I got to the end of the first paragraph before i got bored and stopped reading. You're repeating yourself.
If you can't accept the purpose of the exercise that's fine. You don't have to engage with it or in fact agree with it. Part of getting the right answer is asking the right questions. This was clearly pointed out in the lecture - which i assume you still haven't watched.
This question was based on a similar senirio that was actually used in a court of law to wrongly convict a mother of murdering her 2 children. As the jury, and the defence never bothered to consider the implications of the statistics that were being discussed - they just accepted it - that woman went to jail for years!
Look above your post - how many posters either went for 99%, gave a silly answer or didn't post because they knew by the way that the question was posed that the answer wasn't going to be obvious? That was the point.
In any normal situation if that question was posed in a natural environment, most people would come to the wrong conclusion. Due to the way i posted this, it was obvious that the obvious answer was not the correct one. The entire point of the question is to emphasize how easy it is to draw the wrong conclusion from statistical data presented like this or more importantly, draw conclusions when there is no where near enough information to draw them. Now i've had to spell it out however, it's kind of defeated the purpose....

RJT.

 

[vonQuux]

No, you've yet to hit a nerve. I got to the end of the first paragraph before i got bored and stopped reading. You're repeating yourself.

Pot, meet kettle. Kettle, pot.

If you can't accept the purpose of the exercise that's fine. You don't have to engage with it or in fact agree with it.

You act as if you've got some standing to offer me permission. Good luck with that.

Part of getting the right answer is asking the right questions.

Perhaps but a precursor to asking the right questions is to ask an unambiguous question. If you don't ask a question whose meaning is clear, the questions that result can't be very useful, can they?

This was clearly pointed out in the lecture - which i assume you still haven't watched.

It's amusing that you criticize me for not viewing the lecture while openly admitting to not bothering to read my post.

I didn't watch the lecture because my complaint is about the question, not the answer. I'll ask again; was the purpose of the "statistics puzzle" to ask an ambiguous question?

If this was your goal, it wasn't a statistics puzzle, it was an English syntax parsing puzzle.

If it was not your goal, then you failed.

Pick.

In either case there is no need to watch the lecture.

Look above your post - how many posters either went for 99%, gave a silly answer or didn't post because they knew by the way that the question was posed that the answer wasn't going to be obvious?

I don't accept your conclusions.

It's just as reasonable to believe that people who said "99%" did so because the question lacked the resolution to offer a more precise answer. I'd also argue that those who gave a silly answer -- like me -- did so because the question could not even be agreed upon.

That was the point. In any normal situation if that question was posed in a natural environment, most people would come to the wrong conclusion. Due to the way i posted this, it was obvious that the obvious answer was not the correct one.

How many times do I have to repeat this?

IF YOU POSE A QUESTION POORLY, YOU CANNOT DISTINGUISH BETWEEN A LISTENER WHO JUMPED TO A WRONG CONCLUSION, A LISTENER WHO MADE DUE WITH THE INFORMATION AVAILABLE OR A LISTENER WHO THREW THEIR HANDS UP IN FRUSTRATION AND TOSSED OUT A GOOFY ANSWER.

And, as I've pointed out many, many, many times, YOU'RE NO LONGER ASKING A STATISTICS QUESTION.

The entire point of the question is to emphasize how easy it is to draw the wrong conclusion from statistical data presented like this or more importantly, draw conclusions when there is no where near enough information to draw them.

...and you've done this by drawing the wrong conclusion about the people posting answers and done so without enough information to draw them.

Congratulations. You've made the point but not quite in the way you expected.

You know, you could have saved yourself a lot of trouble by simply admitting you munged the question. There's no dishonor in that. Instead you decided to chase your losses.

Bad call.

vQ

PS: Since you don't read my posts, I trust there will be no reply to this one, right? :laugh:

 

[la_dee_daa]

thanks

RJT thanks it was a great statistics puzzle. For those who don't see the value in how it was posted are missing out on its whole point, end of story.
I enjoyed the video and the points it rasied to.

:cow:

 

[MAZ]

Pot, meet kettle. Kettle, pot.



You act as if you've got some standing to offer me permission. Good luck with that.



Perhaps but a precursor to asking the right questions is to ask an unambiguous question. If you don't ask a question whose meaning is clear, the questions that result can't be very useful, can they?



It's amusing that you criticize me for not viewing the lecture while openly admitting to not bothering to read my post.

I didn't watch the lecture because my complaint is about the question, not the answer. I'll ask again; was the purpose of the "statistics puzzle" to ask an ambiguous question?

If this was your goal, it wasn't a statistics puzzle, it was an English syntax parsing puzzle.

If it was not your goal, then you failed.

Pick.

In either case there is no need to watch the lecture.



I don't accept your conclusions.

It's just as reasonable to believe that people who said "99%" did so because the question lacked the resolution to offer a more precise answer. I'd also argue that those who gave a silly answer -- like me -- did so because the question could not even be agreed upon.



How many times do I have to repeat this?

IF YOU POSE A QUESTION POORLY, YOU CANNOT DISTINGUISH BETWEEN A LISTENER WHO JUMPED TO A WRONG CONCLUSION, A LISTENER WHO MADE DUE WITH THE INFORMATION AVAILABLE OR A LISTENER WHO THREW THEIR HANDS UP IN FRUSTRATION AND TOSSED OUT A GOOFY ANSWER.

And, as I've pointed out many, many, many times, YOU'RE NO LONGER ASKING A STATISTICS QUESTION.



...and you've done this by drawing the wrong conclusion about the people posting answers and done so without enough information to draw them.

Congratulations. You've made the point but not quite in the way you expected.

You know, you could have saved yourself a lot of trouble by simply admitting you munged the question. There's no dishonor in that. Instead you decided to chase your losses.

Bad call.

vQ

PS: Since you don't read my posts, I trust there will be no reply to this one, right? :laugh:

Do you feel better getting all that off your chest? Now have yourself a good cry and settle down. Let me guess, you're a college kid that needs his knowledge spoon fed to him with no ability to obtain answers from questions not directly spelled out for you. Get over it and stop being just another sheep in the flock.

 

[jack.jackson]

I believe the Moral of this puzzle, Demonstrates some simple logic. In order to get a accurate % of something being true, is that any and all factors must Calculated into the Equation.

For example many factors would have to be Calculated, In order to find what a persons chances of having a heart-attack over a year period would be.
Such as Weight, Gender, Age, History, and Life-style Habits. Even Something as I onced Smoked, but longer do, should be factored In.

 

[RJT]

Pot, meet kettle. Kettle, pot.



You act as if you've got some standing to offer me permission. Good luck with that.



Perhaps but a precursor to asking the right questions is to ask an unambiguous question. If you don't ask a question whose meaning is clear, the questions that result can't be very useful, can they?



It's amusing that you criticize me for not viewing the lecture while openly admitting to not bothering to read my post.

I didn't watch the lecture because my complaint is about the question, not the answer. I'll ask again; was the purpose of the "statistics puzzle" to ask an ambiguous question?

If this was your goal, it wasn't a statistics puzzle, it was an English syntax parsing puzzle.

If it was not your goal, then you failed.

Pick.

In either case there is no need to watch the lecture.



I don't accept your conclusions.

It's just as reasonable to believe that people who said "99%" did so because the question lacked the resolution to offer a more precise answer. I'd also argue that those who gave a silly answer -- like me -- did so because the question could not even be agreed upon.



How many times do I have to repeat this?

IF YOU POSE A QUESTION POORLY, YOU CANNOT DISTINGUISH BETWEEN A LISTENER WHO JUMPED TO A WRONG CONCLUSION, A LISTENER WHO MADE DUE WITH THE INFORMATION AVAILABLE OR A LISTENER WHO THREW THEIR HANDS UP IN FRUSTRATION AND TOSSED OUT A GOOFY ANSWER.

And, as I've pointed out many, many, many times, YOU'RE NO LONGER ASKING A STATISTICS QUESTION.



...and you've done this by drawing the wrong conclusion about the people posting answers and done so without enough information to draw them.

Congratulations. You've made the point but not quite in the way you expected.

You know, you could have saved yourself a lot of trouble by simply admitting you munged the question. There's no dishonor in that. Instead you decided to chase your losses.

Bad call.

vQ

PS: Since you don't read my posts, I trust there will be no reply to this one, right? :laugh:

No, you actually got yourself into enough of a tizzy that i read the first couple and last couple of paragraphs this time. When you grow up enough to realize that the world doesn't always make questions easy for you, then i might spend some time conversing with you. Until then you've bored me enough and managed to turn a thread that was fairly simplistic in intention into something hostile and negative.
Oh and by the way, the question was lifted word for word out of the lecture. Maybe if you'd just watched it in the first place you could have avoided all of this but some people just ain't prepard to put any effort in lol.

RJT.

 

[vonQuux]

No, you actually got yourself into enough of a tizzy that i read the first couple and last couple of paragraphs this time.

Nope, no tizzy. I'm just thorough. :grin:

If I was less verbose I'd run the risk of being imprecise which would be kind of ironic. Unfortunately, I guess I haven't been verbose enough. Repeated attempts to outline my argument (in finer and finer increments) have failed to meet any kind of rebuttal.

When you grow up enough to realize that the world doesn't always make questions easy for you, then i might spend some time conversing with you.

Ad hominem.

Until then you've bored me enough and managed to turn a thread that was fairly simplistic in intention into something hostile and negative.

The only person here who's hostile is you. I've done nothing but point out that (a) the question was not a statistics puzzle and (b) the conclusions you reached about the people who posted replies are illogical.

Perhaps you're confusing attacks on your logic with hostility.

Oh and by the way, the question was lifted word for word out of the lecture. Maybe if you'd just watched it in the first place you could have avoided all of this but some people just ain't prepard to put any effort in lol.

That's called "argument from authority."

RJT, in not one of our exchanges have you been able to refute any of my points. Not once. It's not that you tried and failed, you haven't even tried. Saying I'm wrong is not the same as showing why it is wrong.

Instead you're resorted to personal invective. So be it. That reflects on your character, not mine.

But I'll try again, further splitting my argument even smaller bite-sized pieces, to demonstrate that I'm focused on the disagreement and not on you.

Posit: The question, as posed, cannot be agreed upon.

Posit: Confusion over the meaning of the question is distinct from realizing that there are necessarily elements missing from the question.

Posit: Confusion over the meaning of the question precludes a realization that there are necessary elements missing from the question. In other words, it is possible to see that the problem as posed is imprecise without realizing that one needs more data to solve it.

Posit: A person with zero math skills but excellent English syntax parsing skills can see that the question is imprecise.

Conclusion: This is not a statistics puzzle.

It's great that you posted it word for word but if someone cannot defend the most basic premise of the lecture against the simplest of attacks, I would suggest that perhaps they do not understand it very well.

Perhaps I'm wrong. Show me the flaw in my logic.

vQ

 

[RJT]

Perhaps I'm wrong. Show me the flaw in my logic.

vQ

Easy, the flaw in you logic is that you want and won't accept anything less that a solution, when the purpose of the exercise is to realise that a solution cannot be drawn due to lack of information.
When pitched, most people who are asked this question will give the wrong answer. They will assume that they can draw a conclusion that they cannot.
All you needed to do to get this "right" if that's what you so desperately need, was to realise that, despite what might appear an obvious solution, the evidence was inconclusive. No problems with wordings of questions - just simply a puzzle to show how easy it is to draw incorrect conclusion from simple statistics. The purpose of the lecture was to show how easy it is to misinterpret and draw erronious conclusion from even the most basic of stats.
Now you've actually lead me to a point where i'm repeating myself and i'm even boring me. Accept that there is no problem with the question or don't. I really couldn't care less by this point.
I can't actually believe i've wasted more time on this....

RJT.

 

[vonQuux]

Easy, the flaw in you logic is that you want and won't accept anything less that a solution, when the purpose of the exercise is to realise that a solution cannot be drawn due to lack of information. When pitched, most people who are asked this question will give the wrong answer. They will assume that they can draw a conclusion that they cannot. All you needed to do to get this "right" if that's what you so desperately need, was to realise that, despite what might appear an obvious solution, the evidence was inconclusive. No problems with wordings of questions - just simply a puzzle to show how easy it is to draw incorrect conclusion from simple statistics. The purpose of the lecture was to show how easy it is to misinterpret and draw erronious conclusion from even the most basic of stats.

You're going through great lengths to avoid the simple question, even as I've broken it down into sentence-length bits of logic.

I have argued that if a person with no statistics (or even math) skills can logically declare this question unanswerable, it is not a statistics puzzle. I am not arguing that the rest of the lecture has no merit, I am not arguing that there may be statistical elements further on in the lecture that are useful.

I'm arguing the very single, narrow point that the question, as posed, was malformed to the point where the question of statistics became a non sequitur. That is:

non sequitur: noun; a statement (as a response) that does not follow logically from or is not clearly related to anything previously said.

If the title of the post is "statistics puzzle" and if a person with no statistics knowledge can show that the question, as phrased, is unanswerable for reasons that do not rely upon statistics, then the subject of statistics is a non sequitur.

On one hand, it sounds to me like you're vaguely conceding the question, as phrased, was flawed but you can't quite bring yourself to admit it.

If you're not admitting this, which specific line of reasoning is mistaken?

This is how logic works, my friend. It is broken down into discrete elements so the flaws can be laid bare.

But please, let's dispense with the theatrics. You've spent a great deal of time claiming to not read my posts, claiming to read half my posts, expressing boredom a number of times and lobbing insults when all you really ever had to do was put your ego aside and address the question.

vQ

 

[sagefr0g]

......the purpose of the exercise is to realise that a solution cannot be drawn due to lack of information.

wow i got it right! that's exactly what i concluded with in seconds of reading the puzzle. damm. so i just gave up on it. but Sonny's puzzle i did try and guess (lol i wouldn't know the proper reciped to follow for getting the answer)
the thing is i interpreted his problem in the sense that each and every one of the ten thousand player were in fact tested. lol. i don't think thats what his question was getting at. so maybe thats what von is getting at is that to differant minds that have a tendency to work on problems from differant viewpoints such questions can be rather ambiguous.

When pitched, most people who are asked this question will give the wrong answer. They will assume that they can draw a conclusion that they cannot.
All you needed to do to get this "right" if that's what you so desperately need, was to realise that, despite what might appear an obvious solution, the evidence was inconclusive. No problems with wordings of questions - just simply a puzzle to show how easy it is to draw incorrect conclusion from simple statistics. The purpose of the lecture was to show how easy it is to misinterpret and draw erronious conclusion from even the most basic of stats.
.......

RJT.

so maybe one point would be when asked a question one needs to ask one's self questons about the question and hope one can answer at least those lol.

 

[vonQuux]

so maybe thats what von is getting at is that to differant minds that have a tendency to work on problems from differant viewpoints such questions can be rather ambiguous.

In a sense, yes.

We've all seen instances where polls are taken and the questions are worded in such a way that the answers cannot be trusted. For example, it's generally a bad idea to have a poll where one of the available options contains two elements because the reader might agree with one but not the other. When reading the results of such a poll, one cannot know if a person who picked that answer really wanted both or if a person who avoided that answer didn't actually agree with 1/2 of that particular answer.

Similarly, I would suggest that a properly formed question would still demonstrate the statistics element of the puzzle just fine.

Put another way, if the question were merely incomplete, this would be sufficient to demonstrate the point. The introduction of imprecise language muddies the problem rather than clarifying it.

Further, if the question is both incomplete AND ambiguous, one cannot suss out the motives of the people answering the question.

It is in this narrow area that I take umbrage with RJT. He has made assumptions about the mindset of the responders by relying on bad data and refuses to acknowledge that a properly-formed puzzle would be every bit as useful in demonstrating the difficulties of using statistics in the courtroom (or everyday life for that matter).

vQ

 

[RJT]

so maybe thats what von is getting at is that to differant minds that have a tendency to work on problems from differant viewpoints such questions can be rather ambiguous.

Yup, i can see that - yet it still doesn't in anyway invalidate the task. Just because i didn't state right at the start that "the real task is to realise that the qustion has no clear answer without further information", or "what else do you need to know to answer this?" doesn't change a thing.
You are confronted with statistics just as vague and ambiguous every day of your life. In the media, your place of work and your social life and every day you will take for granted what conclusions can be drawn from them - generally the conclusion that the author wishes you to draw.
In the case mentioned above, the jury and the court both accepted the testimony of an "expert" witness who had drawn completely erroneous conclusions simply because an equally ill-founded statistic seemed logical. Nobody questioned this because it seemed intuitive.
The exercise was actually to test whether or not you would question the validity of drawing any conclusion from the statistic. VO can huff and puff all he likes, but if you pick up a new paper right now there's a safe bet that you could find % being used in a similar way.

RJT.