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17th May 2019, 05:11 AM  #1 
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Probability Problem
So I've been going back and forth with a friend of a friend online regarding the Monte Hall Problem. It has devolved into disagreements about probabilities and I'm having a hard time. He has gone into 'actual' and 'perceived' odds. My latest example is 'When you dive by my house, I'm either sitting on the porch or I'm not. What is the probability I am sitting on my porch when I drive by.' He says 'each time there will be a 1 in 2 chance'. Which I kind of get, given there are two possible outcomes (sitting on my porch or not) to one event (him driving by my house). He calls those 'actual' odds while 'perceived' odds can't be calculated until my parch sitting habits are examined. I assume there is some sort of fallacy involved saying out of two options they both have an equal chance of happening, but google is failing me. Does anyone have any good examples that demonstrate this fallacy?

17th May 2019, 05:23 AM  #2 
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The easiest way to illustrate Monty Hall is to say there are a hundred doors, not three.
Monty opens all but two. The chance has clearly switched from 1:100 to 1:2. 
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17th May 2019, 05:25 AM  #3 
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There are several threads dedicated to this problem. I suggest searching them. Rehashing the same arguments can be tiresome.

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17th May 2019, 05:25 AM  #4 
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There being two possible outcomes doesn't necessarily mean those outcomes are equally probable. Roll two dice and you either will or will not roll 66. The chances of rolling 66 are 1 in 36.
If you, however, spend a random 50% of your time on the porch then he'd be right in calling the chances 1 in 2. 
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17th May 2019, 05:27 AM  #5 
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17th May 2019, 05:48 AM  #6 
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17th May 2019, 05:59 AM  #7 
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17th May 2019, 06:08 AM  #8 
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There is either a flying saucer crew from the Omicron Persei system doing an interplanetary survey in my back yard right now, or there isn't. I haven't checked.
Fiftyfifty? 
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17th May 2019, 06:24 AM  #9 
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Well, in fact, there's information there that can be used to form at least a vague estimate of the probabilities involved. Since none of us has ever seen a f.s.c.f.t.O.P.s, and history doesn't record one ever being seen, we would typically form an initial estimate that the odds were vanishingly small. On the other hand, since people are known to sit on porches, an initial estimate might be that the odds against are not too great  unless, of course, SumDooder's house doesn't actually have a porch, in which case we go back to vanishingly small. The point is that probability is about dealing with the level of uncertainty in the truth value of a statement, and it's very difficult to make a statement that conveys no information whatsoever as to its probability.
Dave 
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17th May 2019, 06:27 AM  #10 
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17th May 2019, 06:32 AM  #11 
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Again this is why I think the Three Prisoner Problem demonstrates it better then the Monty Hall Problem.
In the Monty Hall Problem people tend to get hung up on two factors; whether or not Monty is being honest and the fact that the individual can change their choice midway into the problem. The Three Prisoner Problem removes those variables and still retains the same counterintuitive nature. 
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17th May 2019, 06:50 AM  #12 
Penultimate Amazing
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It's just Bayes, isn't it?
You modify your estimate based on other assumptions and new information. Either I'm a dog or I'm not. If you don't know a lot about dogs, you might provisionally put the probability of me being a dog much higher than someone who does know a lot about dogs. And that's okay. 
17th May 2019, 06:54 AM  #13 
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17th May 2019, 06:58 AM  #14 
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Tossing a bent coin. It will either land heads or tails, but the probability is not 0.5 heads and 0.5 tails.
In fact most texts will introduce coin examples with "assume a fair coin". My old uni statistics text calls statistics a "semiempirical" branch of mathematics for this reason. 
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17th May 2019, 07:04 AM  #15 
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It's a relativistic game. The odds are fiddy fiddy for any observer who did not know which door you picked, whether they were there at the beginning or walked in when there were only two left. But you have the additional information of the process of elimination which changes the odds from your perspective.

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17th May 2019, 07:06 AM  #16 
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The nontheoretical character of metaphysics would not be in itself a defect; all arts have this nontheoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it.  Rudolf Carnap "Philosophy and Logical Syntax" 

17th May 2019, 07:44 AM  #17 
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17th May 2019, 08:23 AM  #18 
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Here's a moneymaking opportunity: When you cut a deck of cards, the result will either be the six of spades or not. So, according to the friend, it's 5050. See if you can get him to bet.

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17th May 2019, 08:27 AM  #19 
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 "Ernest Hemingway once wrote that the world is a fine place and worth fighting for. I agree with the second part."  Detective Sommerset  "Stupidity does not cancel out stupidity to yield genius. It breeds like a bucketfull of coked out hamsters."  The Oatmeal  "To the best of my knowledge the only thing philosophy has ever proven is that Descartes could think."  SMBC 

17th May 2019, 08:55 AM  #20 
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His friend isn't stupid, so he probably won't agree to that bet. From the OP:
He calls those 'actual' odds while 'perceived' odds can't be calculated until my parch sitting habits are examined.The friend clearly understands the need to account for additional information when assessing probability. His argument is that when given an binary choice with no additional information, or with information that does not further clarify the probability, the starting estimate is going to be 50/50. Then that estimate will change as more information is added. This seems perfectly cromulent to me. In reality, of course, more information is almost always present  or assumed to be present  at the start anyway. I already know a lot about decks of cards, such as tarot cards, or the kind of playing cards that are customarily used for this kind of activity. So I would not take the bet. Not because the friend's reasoning is flawed, but because the friend's reasoning provides for adjusting the estimated probability based on additional information. Trebuchet's moneymaking "opportunity" rests on taking only the first half of the friend's argument, and ignoring the second. This isn't really a question about probability. It's a question of how we assign likelihood as a cognitive heuristic. What I want to know is, is it even possible to have a "raw" likelihood, without any modifying information? The friend's argument seems correct but largely academic. In the real world, phenomena carry too much baggage to admit a baggagefree starting probability. 
17th May 2019, 08:56 AM  #21 
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I'm not sure that is correct?
For instance, if you write down all the permutations for each door you start with, objectively in 99 of those permutations you end up on a goat and only in 1 permutation you are on a car, so swapping is objectively the correct thing to do. Any observer who walks in at the end and just sees your door and one other, if they think it's 50/50 they're wrong, and it can be demonstrated they're wrong by again just writing down all the permutations, including the observer picking your door 50% of the time and picking the other door 50% of the time. If their odds were indeed 50/50 then they would end on the car half the time, but they obviously won't for the same reason the player doesn't end on the car half the time, but on a goat 99 times out of the 100. 
17th May 2019, 09:17 AM  #22 
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17th May 2019, 09:29 AM  #23 
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We already know that out of the 100 permutations, the player is on a goat 99 times and on the car 1 time.
The observer picks the players door in 50 of those permutations and the other door in the other 50 permutations, it's not possible for the observer to be on the car half the time, when we also know that the observer has picked the players door half the time. 
17th May 2019, 09:34 AM  #24 
Merchant of Doom
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I think you're missing that the newcomer does not know how the other 98 doors were opened. IOW, it could be that the player got 99 picks at doors, and the one he's on now is his last.
The odds only change if you have the knowledge of how the other doors were opened. ETA: Or more accurately, the newcomer can only evaluate the odds on the basis of information he has; if he doesn;t know how the process worked that got them down to 2 doors, he can't evaluate the odds accurately. 
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17th May 2019, 09:35 AM  #25 
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It is not a fallacy. If you have no information about the outcome of a trial whatsoever except that it may or may not happen then it is valid to assign a probability of 0.5 to that outcome.
The thing is that we actually have a lot of information about people sitting on porches and others have mentioned some of the factors that may influence the probability of seeing somebody sitting on a porch. We may not be able to give a reliable estimate of the probability but from what we can deduce, we can reliably say the the probability would be less than 0.5. "Perceived" and "actual" odds is not a thing. What we have is "conditional" probability. That means the probability of an outcome depends (usually) on the conditions of the trial. In fact, there is no such thing as "unconditional" probability. A common condition is that there is a number of outcomes of equal probability and that the outcome is random. In a coin toss for example, we say that heads or tails is equally likely and don't consider things like the coin landing on its edge or snatched midair or being tossed by a cheat etc. 
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17th May 2019, 09:52 AM  #26 
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17th May 2019, 09:55 AM  #27 
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17th May 2019, 10:50 AM  #28 
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There is no chance that you are sitting on your porch if you are driving by, because you are obviously in your car.
As for your problem with the Monte Hall puzzle, the simplest way to prove it's better to switch is to sit down with your friend and play it out a few times. 
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17th May 2019, 11:59 AM  #29 
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What about the box Carol Merrill just brought out?

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17th May 2019, 12:09 PM  #30 
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17th May 2019, 01:49 PM  #31 
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There are 2 ways of understanding probabilities: Frequentist and Bayesian.
The frequentist view is that a probability is information about how often something happens; or the frequency at which something happens. If you told a frequentist that you sit on the porch 1 hour each day, then he would say that the probability of finding you on the porch at any time during the day is 1/24. Without any such information, the frequentist would deny that the question makes sense: You either sit on the porch or you don't. The probability is either 0 or 1 and we simply don't know either way. The Bayesian view is more abstract. Probability is a statement about your knowledge or lack thereof. Under this view you could just pull a number out of your ass. You could say: At this time of day, almost no one sits on the porch so the probability is 5%. Or you could say:He knows I'm coming so he's probably waiting on the porch. The probability is 80%. If you really don't know either way you might as well chose 50/50. Neither number is right or wrong in any mathematical sense. Bayesian statistics provides mathematical methods to refine the initial guess (called prior probability) with empirical data. Every time you drive past the porch and see someone sitting there or not, your guess is nudged closer to the true value, regardless of how right or wrong your initial guess was. How to best come up with a prior is subject to contentious debate. https://en.wikipedia.org/wiki/Prior_...rmative_priors 
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17th May 2019, 09:07 PM  #32 
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19th May 2019, 09:20 AM  #33 
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So your friend is either on the porch or not, so it's 1 of 2.
So your friend is either on the porch, or on the sofa or on neither, so it's 1 of 3. So your friend is either on the porch, or on the sofa or in the kitchen cooking dinner or none of those, so 1 in 4. 
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19th May 2019, 09:37 AM  #34 
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I'm surprised no one has brought up the classic example:
When visiting Egypt, poster shemp is either dryhumping the Sphinx, or he is not. 50/50 odds? 
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19th May 2019, 09:51 AM  #35 
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19th May 2019, 09:54 AM  #36 
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"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent."  Galbraith, 1975 

19th May 2019, 10:04 AM  #37 
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19th May 2019, 10:05 AM  #38 
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That is similar to the fallacy used to prove that it is almost certain that there is life on planet X.
We have no knowledge of whether there are horses on planet X or not so the probability is 50% We have no knowledge of whether there are cows on planet X or not so the probability is 50% So the probability that there are neither horses nor cows on planet X is 50% x 50% = 25%. Add all the other species of life on Earth in the mix and we can conclude that the probability that none of them exists on planet X > 0% so it is almost certain that some form of life on planet X exists. The flaw is that these are all conditional probabilities but we are not expressly told what conditions have been assumed. The starting point is that the probability that life can be supported on planet X is 50%. The probability that life exists on planet X given that life can be supported on planet X is 50% etc etc. 
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