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Probability quiz
Topic Started: May 28 2015, 02:03 PM (544 Views)
Klaus
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HOLY CARP!!!
Imagine that the police has breathalyzers which report false drunkenness in 5% of the cases in which the driver is sober. However, the breathalyzers never fail to detect a truly drunk person. 1/1000 of drivers are driving drunk. Suppose the policemen then stop a driver at random, and force the driver to take a breathalyzer test. It indicates that the driver is drunk. We assume you don't know anything else about him or her. How high is the probability he or she really is drunk?
Trifonov Fleisher Klaus Sokolov Zimmerman
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Mikhailoh
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If you want trouble, find yourself a redhead
That's the wurst quiz I've taken all day.
Once in his life, every man is entitled to fall madly in love with a gorgeous redhead - Lucille Ball
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jon-nyc
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Cheers
1.96270853778214%
In my defense, I was left unsupervised.
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jon-nyc
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Bayes 101.
In my defense, I was left unsupervised.
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Horace
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If all you know is that a randomly selected person has tested as drunk, then they fall into one of two mutually exclusive categories:

1/1000: truly drunk
999/1000 * .05: not drunk

So the answer is the proportion of truly drunk people to the not drunk people, based on those definitions.
As a good person, I implore you to do as I, a good person, do. Be good. Do NOT be bad. If you see bad, end bad. End it in yourself, and end it in others. By any means necessary, the good must conquer the bad. Good people know this. Do you know this? Are you good?
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Horace
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Hmm, my answer turns out to be 2.002002002%. But I just googled it, and Jon's answer is right. Very close answers but I'm not sure why I'm wrong.
As a good person, I implore you to do as I, a good person, do. Be good. Do NOT be bad. If you see bad, end bad. End it in yourself, and end it in others. By any means necessary, the good must conquer the bad. Good people know this. Do you know this? Are you good?
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Horace
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Oh, duh. I was calculating percentages wrong. I was taking the proportion and converting it to percentage by multiplying by 100. But that ain't a percentage. You don't do (a/b) * 100. You do a / (a + b) * 100.

Alert all the fourth grade kids.
As a good person, I implore you to do as I, a good person, do. Be good. Do NOT be bad. If you see bad, end bad. End it in yourself, and end it in others. By any means necessary, the good must conquer the bad. Good people know this. Do you know this? Are you good?
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Klaus
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Bayes is actually not needed to compute the probability in this example.

If you check 1,000 people, 999 will be sober and 1 will be drunk on average.

5% of those 999 are 49.95. Hence the device will report 50.95 drunk persons, only one of them correctly. 1 of 50.95 is the 1.962% Jon mentioned.

The reason why I like the example is that the intuitive guess of most people is way off.
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jon-nyc
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No, you don't need Bayes theorem to solve it, and indeed I just walked through the simple logic Horace laid out.
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taiwan_girl
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Without thinking about it, I would have guessed a lower percentage.
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Moonbat
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Pisa-Carp
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be wakened and interviewed on Monday only. If the coin comes up tails, she will be wakened and interviewed on Monday and Tuesday. In either case, she will be wakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your belief now for the proposition that the coin landed heads?"

What should Beauty answer? (We can interpret 'belief now for the proposition... " to mean "what is the chance that the coin was heads").
Entia non sunt multiplicanda praeter necessitatem
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John D'Oh
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MAMIL
I know what my answer would be, but this is a family forum.
What do you mean "we", have you got a mouse in your pocket?
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jon-nyc
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1/3.
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Klaus
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1/3 sounds plausible, since there are three interview scenarios, and only in one of them was the coin heads.

But I think this is not the answer because the three scenarios are not equally likely. Let me think some more...
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Klaus
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1/2.
Trifonov Fleisher Klaus Sokolov Zimmerman
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Klaus
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<deleted nonsense>
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jon-nyc
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Run the experiment 1000 times. She is awaken 500 times when it's heads and 1000 times when it's tails (you know what I mean).
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Larry
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Mmmmmmm, pie!
Nah. I think if you do her head you'll wake her every time, but you might manage to do her tail without waking her if you're careful...
Of the Pokatwat Tribe

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John D'Oh
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MAMIL
Klaus
May 29 2015, 09:32 AM
<deleted nonsense>
POTD.
What do you mean "we", have you got a mouse in your pocket?
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jon-nyc
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Klaus - looks like there are names for us. You're a halfer and I'm a thirder.

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem
In my defense, I was left unsupervised.
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John D'Oh
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MAMIL
jon-nyc
May 29 2015, 10:05 AM
Klaus - looks like there are names for us.
There certainly are. Do you want to hear mine?
What do you mean "we", have you got a mouse in your pocket?
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Klaus
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HOLY CARP!!!
jon-nyc
May 29 2015, 10:05 AM
Klaus - looks like there are names for us. You're a halfer and I'm a thirder.

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem
Interesting. I found your argument sound, but then again I could not find a flaw in my argument either.
I guess the Wiki page explains why.

But I find it counterintuitive that there is no unique, unambiguous solution [Reading the paper from Elga to find out more].
Trifonov Fleisher Klaus Sokolov Zimmerman
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Horace
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jon-nyc
May 29 2015, 10:05 AM
Klaus - looks like there are names for us. You're a halfer and I'm a thirder.

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem
From the Wiki link:

Quote:
 
The name Sleeping Beauty for the problem was first used in extensive discussion in the Usenet newsgroup rec.puzzles in 1999.


I remember it! I read and sometimes participated in that group at the time. Humbling group of folk. Incredibly good problem solvers.

As for me I'm a thirder. As for halfers, as long as they admit that they'd still be halfers if sleeping beauty was awakened 1000 times rather than twice, I guess they're coherent as far as the argument goes. Difficult to attach any meaning to it though, in that if sleeping beauty were to bet money about heads or tails every time she was wakened, she'd bet on the side that she knew would wake her up more.
As a good person, I implore you to do as I, a good person, do. Be good. Do NOT be bad. If you see bad, end bad. End it in yourself, and end it in others. By any means necessary, the good must conquer the bad. Good people know this. Do you know this? Are you good?
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jon-nyc
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I'll say the halfer position is actually wrong because sleeping beauty does have new information. She knows she was awakened. That means she knows it isn't Tuesday and tails.

If you consider the real question here isnt simply 'what are the odds the coin is heads', rather it's 'what are the odds we wake her up and the coin is heads'. Because that's what the interview exchange represents. Ergo the fact of the interview represents new information to update her priors.

In my defense, I was left unsupervised.
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Klaus
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I think what the puzzle illustrates is that this Bayesian notion of "probability that a hypothesis is true" is somewhat fuzzy.

I tried to solve the puzzle using only maths and basically disregarding intuition, because intuition is often misleading in such cases. And then I ended up with 1/2.

But when I enable intuition, then I find the 1/3 argument more convincing.
Edited by Klaus, May 30 2015, 02:53 AM.
Trifonov Fleisher Klaus Sokolov Zimmerman
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