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Coin tossing, probability and some logic, kinda


Swan Red

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The maths isn't exact, but I think it would be something to the effect of...

 

If you take 1,000 coin tosses as a series, if the coin does have an even chance of heads/tails, in something like 95% of series, the amount of heads will be somewhere between 45% and 55% (this being the normal distribution). The more tosses you have in each series, the more you can narrow down the %ages (so if each series is 10,000 tosses, and the coin is fair, you would expect hears between 48% and 52% of the time).

 

Similarly, the more series of tosses you have, the more confident you can be in the result (i.e. 1,000 series of 1,000 tosses where 95% fall into the normal distribution will give you a higher level of statistical confidence than just 20 series). You need to have lots more series, because random factors could mean a series where every one of the 1,000 coin tosses is heads, but the coin still isn't bias.

 

Using the example of 1,000 tosses per series, if in only 60% of series fall within the normal distribution (after 1,000 series), and heads tends to account for between 35% and 45% within each series, you could conclude with a very high level of confidence that the coin is not a fair coin, and has bias towards tail.

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I'll be asking for an idiots guide to standard deviation soon.

 

One of the things that struck me when looking at some probability theory is the base rate fallacy. I'm probably late to this as well but I'll post it an see.

 

A condition affects 1 in 1000 people, there is a test that gives accurate results 99% of the time you are tested for it and return a positive result. What are the chances you have the condition. Now this presentation is flawed because i'm not distinguishing the error rate for positive results from negative results but most people intuitively give an answer close to 99% where the actual result is closer to 10%. seems really obvious when it was explained but it was far from obvious when I first learned of it.

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1/1000

 

Having a test or not having a test doesn't affect the actual number of people with the condition - which is already stated as 1/1000

 

The only thing that is affected is the perception of the people that have the condition - which we already know isn't 100% accurate.

 

I think you've slightly misunderstood.

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1/1000

 

Having a test or not having a test doesn't affect the actual number of people with the condition - which is already stated as 1/1000

 

The only thing that is affected is the perception of the people that have the condition - which we already know isn't 100% accurate.

 

My bad I failed to explain it adequately.

 

What is the probability you have the condition in the event that you get a positive result. This isn't 1/1000.

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My bad I failed to explain it adequately.

 

What is the probability you have the condition in the event that you get a positive result. This isn't 1/1000.

 

 

Does the frequency of the condition occuring in the population have anything to do with the reliability of the tests? Seems to me they're not connected.

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Does the frequency of the condition occuring in the population have anything to do with the reliability of the tests? Seems to me they're not connected.

 

No but the probability that a person has the condition is dependent on the frequency it occurs in the population.

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My bad I failed to explain it adequately.

 

What is the probability you have the condition in the event that you get a positive result. This isn't 1/1000.

 

 

Having or not having the condition is a binary constant. True or false.

 

99% of the cases showing true would be correct

 

1% of the cases showing true may or may not be correct - what is the exact criteria of that 1%? Are they always assumed to be incorrect? Or is that 1% subject to randomness - a mixutre of correct and incorrect diagnosis?

 

1/1000 would be the number if there was a 100% accuracy in the test

 

The probability you have the condition is still 1/1000 - but the perceived probability is (relying on test results) more than 1/1000 as some of those 'true' results would be wrong. But that number might go up or down depending on the criteria of the wrong results - are they always positively wrong or negatively wrong or randomly wrong?

Edited by Andy @ Allerton
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Yeah I think you've slightly misunderstood

 

1/1000 would be the number if there was a 100% accuracy in the test

 

No, there being a 1/1000 chance is the prior probability, if it is also the posterior probability then the test actually tells us nothing. This is wrong Andy.

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So I didn't set it up as clearly as I might but some numbers.

 

1 in 1,000 people are affected by some condition or 0.1%

1% of the tests will give a false verdict.

 

100,000 people are tested for the condition.

 

100 of these will have it or 1/1000.

1,000 of the people tested will receive inaccurate results.

 

You are still no more than 10% to have the condition if you test positive for it.

 

This can change if the %'s vary for false positives and false negatives but false negatives are expected to be extremely rare given the base rate. For instance there are 10x the number of false results than there are actual cases of the condition.

Edited by Swan Red
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So I didn't set it up as clearly as I might but some numbers.

 

1 in 1,000 people are affected by some condition or 0.1%

1% of the tests will give a false verdict.

 

100,000 people are tested for the condition.

 

100 of these will have it or 1/1000.

1,000 of the people tested will receive inaccurate results.

 

You are still no more than 10% to have the condition if you test positive for it.

 

This can change if the %'s vary for false positives and false negatives but false negatives are expected to be extremely rare given the base rate. For instance there are 10x the number of false results than there are actual cases of the condition.

 

 

Yeah that makes sense because 1% is much higher than 0.1%

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So I didn't set it up as clearly as I might but some numbers.

 

1 in 1,000 people are affected by some condition or 0.1%

1% of the tests will give a false verdict.

 

100,000 people are tested for the condition.

 

100 of these will have it or 1/1000.

1,000 of the people tested will receive inaccurate results.

 

You are still no more than 10% to have the condition if you test positive for it.

 

This can change if the %'s vary for false positives and false negatives but false negatives are expected to be extremely rare given the base rate. For instance there are 10x the number of false results than there are actual cases of the condition.

 

 

Read a book not so long ago that you would probably like - called "The Drunkard's Walk: How Randomness Rules Our Lives" - includes a fair amount of stuff like this re how often we completely misinterpret stats, and of course about how we don't understand the role that randomness plays in probability...

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Read a book not so long ago that you would probably like - called "The Drunkard's Walk: How Randomness Rules Our Lives" - includes a fair amount of stuff like this re how often we completely misinterpret stats, and of course about how we don't understand the role that randomness plays in probability...

I was reading that some years ago, and, pleasingly randomly, lost it (the book, that is). Couldn't remember the title even, so never got round to looking it up to buy it again and finish it - which I now can.

 

Doesn't it cover how Traffic Queues form as well, for no apparent reason, in quite a level of (to me) interesting detail? Starts from the Thermodynamics of superstable systems or such like?

 

* Quite like the logic behind how many people you need in a group to be effectively certain that 2 people will share the same birthday. Probably a total yawn for some of the stattos contributing here, but I never really got past the 'take your own bomb on a plane, then' level of statistical argument when I had to do it at Uni.

Edited by JRC
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So I didn't set it up as clearly as I might but some numbers.

 

1 in 1,000 people are affected by some condition or 0.1%

1% of the tests will give a false verdict.

 

100,000 people are tested for the condition.

 

100 of these will have it or 1/1000.

1,000 of the people tested will receive inaccurate results.

 

You are still no more than 10% to have the condition if you test positive for it.

 

This can change if the %'s vary for false positives and false negatives but false negatives are expected to be extremely rare given the base rate. For instance there are 10x the number of false results than there are actual cases of the condition.

 

probably slightly easier to follow if you'd used 1% & 0.2% or 2% & 0.1%, but i see your point and it is counterintuitive

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Read a book not so long ago that you would probably like - called "The Drunkard's Walk: How Randomness Rules Our Lives" - includes a fair amount of stuff like this re how often we completely misinterpret stats, and of course about how we don't understand the role that randomness plays in probability...

 

Nice one cheers.

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I was reading that some years ago, and, pleasingly randomly, lost it (the book, that is). Couldn't remember the title even, so never got round to looking it up to buy it again and finish it - which I now can.

 

Doesn't it cover how Traffic Queues form as well, for no apparent reason, in quite a level of (to me) interesting detail? Starts from the Thermodynamics of superstable systems or such like?

 

* Quite like the logic behind how many people you need in a group to be effectively certain that 2 people will share the same birthday. Probably a total yawn for some of the stattos contributing here, but I never really got past the 'take your own bomb on a plane, then' level of statistical argument when I had to do it at Uni.

 

hmmm... I don't recall it having anything to do with how traffic forms (which I know is something of a weird mystery) ... Think the Birthday one may be in there though. The book also introduced me to Pascal's triangle, which is something I've ended up using quite a bit.

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hmmm... I don't recall it having anything to do with how traffic forms (which I know is something of a weird mystery) ... Think the Birthday one may be in there though. The book also introduced me to Pascal's triangle, which is something I've ended up using quite a bit.

 

 

Read something about traffic - which is why they are introducing the systems to limit speed staged across several zones - which speeds everything up. It eliminates the concertina effect (or accordion effect if you prefer)

Edited by Andy @ Allerton
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