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2nd February 2010, 04:43 PM | #41 |
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Bingo. My thoughts too.
I was intrigued enough to have a hunt around the web to see what might offer some hints on what the book said. There were a lot of interviews with the author that were quite interesting, focussing on our brain's clumsy handling of statistics. Lots of Monty Hall problem stuff, etc. The closest I could find were some explanations on how it was our ignorance on a coin flip that produced the 50% accuracy, and the huge variation in contributing factors that produced the 50% probability. In other words, if we knew prior to a coin flip all of the factors that went into determining its spin, we'd guess better than 50%. And if we could control all of those factors, we'd be able to vary the probability by more than 50%. At least, that's how I interpreted it. But I could find no figures of 60%, or any hint on how we might consistently get that figure of guessing. It makes no sense. Athon |
2nd February 2010, 05:13 PM | #42 |
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On closer inspection, this is nonsense
I think I had in mind the fact that in a random walk, the distance from the origin will cross any given threshold an infinite number of times. Sadly this does not apply to the hits/trials ratio, as far as I can see. I was tired. Honest |
2nd February 2010, 05:59 PM | #43 |
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2nd February 2010, 06:28 PM | #44 |
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2nd February 2010, 06:37 PM | #45 |
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2nd February 2010, 07:05 PM | #46 |
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2nd February 2010, 07:09 PM | #47 |
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3rd February 2010, 05:44 AM | #48 |
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Of course, you can eventually beat a coin-toss game, as long as you have unlimited resources: http://en.wikipedia.org/wiki/Martingale_(betting_system) Then again, if I had unlimited resources, I could probably think of better things to do. |
3rd February 2010, 05:54 AM | #49 |
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Perhaps what we have here is a garbled version of the fact that sometimes the event is not as random as people assume it is. For example: when people guess if a coin lands "heads" or "tails", they tend to alternate between "heads" and "tails" pretty regularly, rarely guessing the same result more than twice. If the coin-tosser whose tosses someone guesses is human, they, too, are likely (if they get, say, four "heads" in a row) to ignore the last flip and try again, thinking that the row of "heads" shows those last few tosses were somehow not "really" random. This has the joint effect of making correct guesses more likely. But we need to know the context.
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3rd February 2010, 06:01 AM | #50 |
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Welcome CC, I did a ten minute Google search. Your prof has access to paper no one else is talking about.
I think a citation is in order, Author, and date would be nice. The only issue I found on Google is the well know fact that coins tosses are not random, unless they are controlled for. |
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3rd February 2010, 06:08 AM | #51 |
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I've searched through Drunkard's Walk and can't figure out what he is referring to. His point about separating the probability of an event and the probability of a prediction is valid, but it doesn't alter your accuracy in the absence of knowledge, except through chance.
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3rd February 2010, 07:15 AM | #52 |
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Nope. I made no mistake. Volunteering extra information doesn't actually change the probabilities.
For instance, if one was a red coin, and one was a blue coin, and I asked if one came up tails, if you told me 'the blue one came up tails' then the probabilities still stand. Order of flipping is about as relevant as the color of the coin - not at all. You might think that I've learned extra information about the red coin (that it has a 67% probability of being heads), and that that violates the fact that the coin flip is 50/50, but in fact I have not. The reason is that when I ask the question, the coin flip is no longer 50/50 (it is not half way between heads or tails. It is merely one or the other). To put it in populist terms, we've 'looked in the box.' Unless in you're hypothetical you wouldn't tell me if the second one came up tails, but that just makes the entire thing worthless (if we assume you're lying to me, anything goes). |
3rd February 2010, 07:15 AM | #53 |
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Heads er tails?
Very true. It is due to the wisdom of coinage:
http://www.scribd.com/doc/13366448/T...lay-in-One-Act |
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3rd February 2010, 07:39 AM | #54 |
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In order to test this for a coin flip I tried entering 100 guesses of one and zero down a spreadsheet and then used the rand function in the next column to put in 1's and 0's at random, then in the next column I put a one or a zero depending on whether my guess is right.
Then I recalculated it twenty times - I get an average of 49.5 right guesses Tried again with 100 new guesses - after recalculating twenty times an average of 50.1 Tried again with 100 new guesses - and after recalculating twenty times I get an average of 49.2 So it seems to me that the prof is a little off Even if your guesses are not at random, there is still just a 50% chance of each being correct. |
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3rd February 2010, 08:00 AM | #55 |
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No.
Let's look at the possibilities. The blue coin can be heads or tails (B+ or B-) and the red one can be heads or tails (R+ or R-). Possible outcomes are B+R+, B+R-, B-R+, B-R-. You ask, in effect, "Can we eliminate B+R+?". You receive the reply, "We can eliminate B+R+ and B+R-". It doesn't matter that this is not a reply to the question you asked; you have been given information to the effect that two possibilities are excluded. Therefore, the only possible results are B-R+ and B-R-; the probability of red being heads is 50%. The same is true, of course, if you don't even ask the question. In fact, your asking is completely irrelevant to the probability of a particular outcome, as it gives you no information. It's only the information you receive that matters. For the same reason, if I flip two coins, then tell you 'At least one of these coins came up heads. What's the probability of one being tails?' the answer is 2/3, whereas if I tell you 'The first one came up heads [...]' or even 'The one that landed nearer my feet came up heads [...]', then the answer is 1/2. Counter-intuitive but true. Dave |
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3rd February 2010, 08:05 AM | #56 |
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3rd February 2010, 08:15 AM | #57 |
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3rd February 2010, 09:09 AM | #58 |
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Done and done
Okay. A simple play about flipping coins and about how you have a 1 in 16 chance of calling heads, say, four times in a row but that does not mean that the next throw has only a 1 in 32 chance of being heads again.
PEABODY You know what is the most beautiful thing about a coin? It has two sides and only two sides. You got your head side and your tail side, and every time it lands, it lands either heads or tails. It is a piece of the finest craftsmanship. Pleasing to the eye. Durable. Manufactured by the United States Government to exact specifications. It is the gold standard. Not just in commerce, but in a game of chance. You see? and GILLY You damn fool! A coin has got two sides. Only two sides. It don’t remember. It doesn’t know how it landed in the past. Every time it goes up it can only come down heads or tails. Fifty-fifty, Vernon. You bet one hundred and seventy-three dollars against sixty on an even chance. |
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3rd February 2010, 09:18 AM | #59 |
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3rd February 2010, 10:40 AM | #60 |
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That is an interesting challenge and introducing the machine or computer into it allows you to play with the idea. All spreadsheets have a random number generator function. Set up two columns One as the event and One as the prediction. Then compatre the two with an IF statement and them total the results.
I collected data on 20 trials of 100 events each (total of 2000 events). For teh 20 sets of 100 event the accuracy ranged from 41% to 55% correct. With the total average over the 2000 events of 49.3% correct predictions. It would be possible to get a run of 100 that could be 60% but that goes away as you increase the number of trials. Done - 10 minutes - could it be that simple or am I missing something that allows it to be the quoted 57% value. Is that assuming bias (intentional or not) on the part of the participants or equipment (unbalance coin, consistent flipping method). |
3rd February 2010, 11:00 AM | #61 |
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Bringing in 'if one is head guess tails' skews the results if both are tails.
H-H - wrong H-T - Guess right T-H - Guess Right T-T - No guess gimmie So 75% accuracy can be achieved if you require that head be declared if one exists Likewise 66% accuracy can be achieved if you toss out all TT results and require declaration of at least on head |
3rd February 2010, 12:50 PM | #62 |
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I'm not sure I follow that, could you please explain? The reason I ask is that I worked in the casino biz for 10 yrs. and if you get 10 red spins of a roulette wheel everybody is going to bet on black and more often than not, within the next couple, three spins, it will come up black most of the time.
I know in theory if you flip a coin to infinity you can get a million heads in a row but in all my dealing days I never saw 30+ wins either way. There always seemed to me to be what I would call the "odds within the odds", if that makes sense. |
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3rd February 2010, 01:44 PM | #63 |
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The thing is, by the time you're on 10 spins of red, in the next three spins your chances of getting all red are... 1 in 8.
1 in 8 isn't very high. I mean it's not unnoticeable, but it's not very high. So sure, most of the time if you have 10 spins of red, 7/8 of the time at least one of the next three spins will be black (this is ignoring the 0, yes). A run of 30 spins of red will only happen one in one billion run of 30 spins. The probability goes up when the run is longer (as any group of 30 red spins in a row works) but your chances of seeing it in any given night are probably only one in tens of millions. But when you've spun 29 red spins in a row, the chances that the wheel will come up red on the 30th spin are 1/2 (approximate, I do know about 0). Infinity is a tad different. The point is not that you will get 'a million spins of red in a row if you spin infinite times.' The point is you will get a run of one million reds in a row. Exactly. And you will get a run of 1 million, five hundred thousand reds in a row. Exactly. And you will get a run twenty billion spins long where every spin goes red, black, red, black, red, black. Exactly. Every single combination of spins you can possibly imagine, and every one you can't occurs in an infinite string of spins. |
3rd February 2010, 01:55 PM | #64 |
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if you want further data I just slapped together a quick program to test this.
It randomly predicts a number of tosses, then it actually make the tosses and then compares the predictions and the results. I set it to make 100000 tosses 10 times and this is the result I got: 1. 50379 correct predictions. 2. 50135 correct predictions. 3. 49679 correct predictions. 4. 49935 correct predictions. 5. 50026 correct predictions. 6. 49911 correct predictions. 7. 50152 correct predictions. 8. 50063 correct predictions. 9. 49934 correct predictions. 10. 49887 correct predictions. As anyone can see all the outcomes are very close to 50% correct predictions. Your professor should rethink his claim. |
3rd February 2010, 02:35 PM | #65 |
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However your program is randomly generating the guesses and human guesses do not come at random, so it is not really testing the professor's claim.
The one I did about (http://www.internationalskeptics.com...70#post5580170) was based on actual guesses I made, but it still came up with around 50% correct guesses even with a human guesser. |
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3rd February 2010, 02:51 PM | #66 |
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I've just been thinking it through, and we're using the wrong percentages. It should be 75%, not 67%.
We've been assuming the possible outcomes are: HH HT TH Which is true, but HH is twice as likely than either HT or TH alone, so we should be basing our assumptions on the possible outcomes of: HH HH HT TH The reason for this is: We have a 50% chance of getting a coin with a 100% chance of being heads (because that's the coin that the coin-flipper was talking about when he told us that at least one coin came up heads). This gives us a 50% chance of getting heads to start with. We also have a 50% chance of getting a coin with a 50% chance of being heads, which adds another 25% chance of getting heads to the outcome, bringing it up to 75%. If you draw a probability tree, you've gotta keep track of the probability of each branch. Does anyone know if there is any kind of two-coin-flip situation that actually gives you a 67% (or 66.66667%) chance of being right? |
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3rd February 2010, 06:11 PM | #67 |
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No wait... I'm wrong. It is 66.6667% not 75%. Should have thought it through a bit more.
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3rd February 2010, 08:18 PM | #68 |
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I can predict 57% of the time whether or not a pair of dice will show a total greater than seven.
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3rd February 2010, 08:30 PM | #69 |
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3rd February 2010, 09:18 PM | #70 |
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Was the professor in the OP lecturing on statistics in the context of interpreting psychological experiments? Or was he lecturing on the psychology of predictions?
If it's the former then either he wasn't communicating some missing bit of info, or he's just dead wrong - claiming the same probability for coins and cards is enough to destroy the pure statistics bit. I _might_ buy the hypothetical claim that if Person A predicted a coin toss, then tossed a coin without showing the outcome to Person B, then Person B has a roughly 60% chance of "predicting" that Person A's prediction was correct - just by observing Person A's reaction. But that would be a psychological experiment, not a statement on probability. I'm curious if your professor can come up with the original cite. |
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4th February 2010, 12:05 AM | #71 |
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Should this be merged with the thread discussing this in Science, mathematics, etc?
http://www.internationalskeptics.com...d.php?t=166405 A |
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4th February 2010, 06:10 AM | #72 |
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4th February 2010, 07:22 AM | #73 |
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Update - I spoke to the professor again, and he said the coin tosses didn't really work in the context of the theorem he had in mind, but his assertion about Zenner Cards was still valid. He said an average person would accept someone as being psychic if that person could correctly predict the identity of a Zenner Card 4 out of 10 times, but you can do better than that by chance alone (again, up to 57-60%).
I would be surprised if an average person would be "impressed" by 4/10, but I suppose a competent performer could make it seem paranormal. Statistically, the probability of hitting 4 or more out of 10 is about 12% - improbable, but not very impressive. 5 / 10 would be more impressive at 3%, while 6 / 10 would occur roughly .6% of the time. I can't see any validity in the claim that a person could consistently perform better than 40% of the time predicting Zenner Cards. It would be even more difficult to get 40 / 100 (4E-4%) or 400/1000 (which the binomial probability calculator I'm using bugs out on). At those numbers, even 40% would be very impressive, let alone 57-60%. My professor is going to look up the study and get back to me - I've got to imagine the claim it actually makes is different, or else it's woo. |
4th February 2010, 07:55 AM | #74 |
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4/10, assuming traditional decks, and with a sufficient level of repeatability and oversight to assure no trickery, would be sufficient to prove paranormal powers.
Granted, it would probably take 200 or more repetitions, but if you're consistently hitting 40% when you should be hitting 20%, that's insanely good. 60% on zenner cards would just be ridiculous. No joke, if you could improve the odds that much, you'd be psychic (once again, with no trickery. Obviously you can do most anything if you're tricking people, or reading them, or whatever). |
4th February 2010, 08:13 AM | #75 |
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Then I wonder if he's talking about Zener cards which aren't drawn randomly. Uri Geller would ask his audience to pick one of the five cards and then he would guess which one they had picked. Since most people (maybe 60 percent?) would pick the star, all he had to do was guess the star and he would be right for the majority of the audience members.
Otherwise, he may be talking about Zener cards experiments where reasons were found to exclude wrong results (the subject was tired or not concentrating, the examiner was a goat, etc.), which then biased the total results to an overall hit rate that was greater than chance (although I don't think it was as high as 57 percent. We should probably just shut-up and wait for him to look up the study. Linda |
4th February 2010, 08:39 AM | #76 |
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4th February 2010, 08:41 AM | #77 |
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Let me just think out loud for a moment about the cards.
You have a set of five cards. One is drawn, and you make a guess. Your chance of getting it right is 0.2. A second is drawn; if you know what the first one was, your chance of predicting the second is 0.25. Your chance of getting the third is 0.3º, the fourth is 0.5 and the fifth 1 (because you can eliminate the other four). I haven't done the arithmetic, and I'm not sure I can be bothered, but it seems to me that the expectation value is that you'll get about three right purely by guess and elimination. Is that what he meant? Dave |
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4th February 2010, 08:52 AM | #78 |
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I suck at statistics but I'm a decent programmer. I like to write simulations of things like this. Yes, computers are only pseudo-random but I think it's close enough.
10 trials of 100,000 flips each - random head/tail generated as prediction, then random head/tail for the flip Trial # 01 100,000 flips 50.14% correct - 50.21% heads 49.79% tails Trial # 02 100,000 flips 49.90% correct - 49.94% heads 50.06% tails Trial # 03 100,000 flips 50.17% correct - 49.96% heads 50.04% tails Trial # 04 100,000 flips 50.00% correct - 50.02% heads 49.98% tails Trial # 05 100,000 flips 49.77% correct - 49.94% heads 50.06% tails Trial # 06 100,000 flips 50.14% correct - 49.93% heads 50.07% tails Trial # 07 100,000 flips 50.40% correct - 50.01% heads 49.99% tails Trial # 08 100,000 flips 50.11% correct - 50.09% heads 49.91% tails Trial # 09 100,000 flips 49.78% correct - 49.89% heads 50.12% tails Trial # 10 100,000 flips 49.97% correct - 50.04% heads 49.96% tails Just to show how sample size matters, here is the same program with trials of only 100 flips. Trial # 01 100 flips 55.00% correct - 57.00% heads 43.00% tails Trial # 02 100 flips 49.00% correct - 48.00% heads 52.00% tails Trial # 03 100 flips 58.00% correct - 50.00% heads 50.00% tails Trial # 04 100 flips 46.00% correct - 47.00% heads 53.00% tails Trial # 05 100 flips 50.00% correct - 53.00% heads 47.00% tails Trial # 06 100 flips 48.00% correct - 51.00% heads 49.00% tails Trial # 07 100 flips 52.00% correct - 51.00% heads 49.00% tails Trial # 08 100 flips 43.00% correct - 55.00% heads 45.00% tails Trial # 09 100 flips 44.00% correct - 48.00% heads 52.00% tails Trial # 10 100 flips 48.00% correct - 47.00% heads 53.00% tails |
4th February 2010, 03:25 PM | #79 |
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That's a damn interesting question. I know I tend to pick heads, and even have a serious case of confirmation bias towards 'always' having coins land on this side.
I try hard to shake it by paying attention to those occasions when it does come up tails, but it's a fairly stubborn feeling. It's possible that I am a statistical anomaly, and I do experience an abnormally large sample of flips coming up head-side. But I tend to keep it in mind as an example of the power of confirmation bias in my own mind. Athon |
4th February 2010, 03:48 PM | #80 |
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Well, the study seems elusive - now several classmates are insisting to have seen it, but no one can give me a link, name, or citation. I'll keep trying to track it down and let you know if I have success.
One of my friends is making the argument that all coin tosses are biased (he says there is no such thing as a "fair coin toss"), and humans can recognize the inherent patterns and use this information to increase their accuracy. I can only buy this to a very limited extent - in a decently controlled experiment, I'd expect any bias to be a fraction of a percent if present at all. At such a level, I would expect most people to be unable to catch on and use this to their advantage. I suppose, however, that it would be possible for someone to use this bias to increase their performance over a long run of trials, but only to a very limited extent (a fraction of 1%, maybe?). I don't see any way to get the 7% improvement forecasted by the alleged study. Perhaps someone more knowledgeable can shed some light on the bias of coin tosses. I'm taking it for granted we're talking a "fair coin," with no special flipping techniques or other funny business. I don't doubt that it's possible to flip a coin in such a way as to make one side significantly more likely to appear, but my prof and classmates have explicitly denied this possibility. When you get into that sort of manipulation, you can make the statistics say anything, but unless I'm seriously mistaken, sufficiently large runs of "fair coin tosses" will always average around 50%. Quite honestly, I'm at a loss to deal with the claims my prof and classmates are making. The only thing that came even close to making sense was the point about bias, but that was grossly exaggerated. Am I crazy, or am I just surrounded by people who have no clue about statistics? ETA: Just looked up statistics on the bias associated with tossing coins:
Quote:
http://www-stat.stanford.edu/~cgates...s/thinking.pdf Absolute best case scenario (and I'm talking biased flips and a participant with the ability to keep track of trials as a professional card counter would), you can increase your accuracy by half a percent. |
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