Statistics puzzle - answers by Friday

RJT

Well-Known Member
#1
Here's an interesting little puzzle for everyone to have a hard think about.
I will post a solution to this on Friday, until then if you genuinely already know the answer, please keep it to yourself and let others have a go -

Suppose we have a diagnostic test for a particular disease which is 99% accurate.
A person is picked at random and tested for the disease.
The test gives a positive result. What is the chance that the person actually has the disease?

RJT.
 

Guynoire

Well-Known Member
#3
I think the puzzle is missing some information. Don't you need to know the disease percentage of the total population to answer this?
 

Sonny

Well-Known Member
#5
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-
 

sagefr0g

Well-Known Member
#6
Sonny said:
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'm just gonna guess for the fun of it.
it'll find 99 ploppies to be card counters and 1 card counter will be correctlly identified. :rolleyes:
edit: erhh that's one out of a hundred a 0.01 probability.
 
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godeem23

Well-Known Member
#7
Sonny said:
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 think this needs to be reworded. The question can be interpreted in the following ways:
1. A person was chosen randomly and identified as a counter. What is the probability that they are actually a counter?
2. A person was random chosen and tested by the software package. What is the probability that the person was BOTH a counter AND identified as a counter?
 

vonQuux

Well-Known Member
#8
Guynoire said:
I think the puzzle is missing some information. Don't you need to know the disease percentage of the total population to answer this?
I was thinking the exact same thing.

vQ
 

blackchipjim

Well-Known Member
#11
puzzle puzzle

I have to agree with the others as far as the missing factors in the equation. Hazarding a geuss with unknown factors would be the same as hitting the state lottery or the mega bucks jackpot. I would have to say the odds would be the same. Taken into account the unknowns I would say the odds of being picked and testing positive would be one in 19million. blackchipjim
 
#12
I dunno but

I think there is still information missing from both the disease hypthetical and the card couting hypothetical--
We don't know whether the term "accurate" applies to FALSE POSITIVES or FALSE NEGATIVES.
You can imagine a test that is 99% accurate in the sense of rarely giving a FALSE POSITIVE (so if it says you have the disease, you probably do), but which is 99.9% INACURATE in giving FALSE NEGATIVES (so that a negative result says just about nothing).
I guess that if the test if 99% accurate in the sense of giving a false positive only 1% of the time, then a postive test result should mean that the subject has a 99% probability of actually having the disease, +/- the margin of error.
If the test is 99% accurate in the sense of giving a false negative only 1% of the time, then the question can't be answered without also knowing the prevalance of the condition (which we don't know in the disease hypo, but we do in the counter hypo).
 

Sonny

Well-Known Member
#13
godeem23 said:
I think this needs to be reworded. The question can be interpreted in the following ways:
1. A person was chosen randomly and identified as a counter. What is the probability that they are actually a counter?
2. A person was random chosen and tested by the software package. What is the probability that the person was BOTH a counter AND identified as a counter?
Indeed the question could be taken either way. Question #1 is what I intended to ask while question #2 is what RJT seemed to be asking. How about a Daily Double? Anyone know the answer to both?

-Sonny-
 

rukus

Well-Known Member
#16
using sonnys example (which has all info you need to answer questions 1 or 2 that were clarified in the followed up) and assuming "correctly identify" means that a counter is identified as a counter 99/100 times and a ploppy is identified as a ploppy 99/100 times, the answers should be as follows:

1. A person was chosen randomly and identified as a counter. What is the probability that they are actually a counter?

> 0.01*0.99/(0.01*0.99+0.99*0.01) = 0.5 => 50% chance of actually being a counter

2. A person was random chosen and tested by the software package. What is the probability that the person was BOTH a counter AND identified as a counter?

> 0.01*0.99 = 0.0099 => 0.99% chance of selection being a counter and identified as such

not sure if Sonny realized the nice symmetry he gave the numbers in the problem such that no calculators are needed. (if so, very nice.)

wow, im really bored at work today.

EDIT: for those interested in the logic/math, check out Bayes Theorem -
http://en.wikipedia.org/wiki/Bayes'_theorem
 
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RJT

Well-Known Member
#17
As promised here is the solution:-

http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool_juries

I'm pleased to see that some of you got it right, although it is somewhat of a trick question.
I'm not going to tell you where exactly you'll find the answer, but it's a very short lecture, about 20min, and i think most of you will find this very interesting.

If you consider yourself at all academic, or simply enjoy learning about just about anything imaginable this website is a fantastic resource, one that i personally have spent many hours of my life trailing though.
This is a bit of an advertisement and i hope the mods don't mind, i don't work for ted or have any connection to them other than i've watched loads of the lectures they've put up.

RJT.
 

Frankie

Well-Known Member
#18
1. I don't have time to watch a 20 min video to get an answer to a problem I can easily solve on my own

2. As several people pointed out, your question lacked several necessary facts to get an answer.
 

RJT

Well-Known Member
#20
vonQuux said:
With all due respect, a precursor to being interested in the answer is probably getting at least two people to agree on the question. :laugh:

Again, not intended as a flame but...

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

RJT.
 
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