Dr. Ted Hill, a PhD professor of mathematics used to divide his class in half. Half the class was asked to go home and flip a ';fair'; coin 200 times. They would write on their paper HHTTHHHTT.. depending on if they got heads or tails. The other class was told to write a string of H's and T's randomly. The professor could tell with a high degree of accuracy which papers were faked quickly. How did he do it?
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I am trying to avoid wildly speculative answer so:
(1) The coins are ';fair'; so they are not weighted to one side or the other so that they give even odds. Most average coins can be considered to be fair.
(2) The decision is made relatively quickly. The professor does not have time to add up all the H's and T's on each paper.
(3) No one is stupid enough to write HTHTHTHTHTHTHT.. 200 times.
- The answer says a lot about our expectations about gambling.Coin tossing (True nature of random outcomes)?
You are trying to avoid wildly speculative answers, but I will go ahead and make and a** of myself, and speculate wildly.
I think the reason for his high degree of accuracy at figuring out which papers were faked and which were not is because we humans do underestimate the power of randomness. You are entirely correct in stating that no one is going to be stupid enough to write HTHTHTHT... (althought I must say I know some folks who are so unimaginative that they might just do that...**sigh**...I know some REALLY stupid people). However, what I think people might very well underestimate is the consecutive number of times a coin toss will result in either heads or tails. I think a person would say, ';Well, I can imagine a coin landing heads up 5 times in a row, and I can imagine it landing heads up 10 times in a row, but when it comes to 15 times in a row, that stretches my imagination, so the longest string of Hs I will put on my paper is 14.'; The fact of the matter is that if you are doing 200 coin tosses, it is entirely possible that you could have a run of either heads or tails which is much longer than 15. Just because the law of averages dictates that your distribution of heads and tails should be near 50/50 doesn't mean that it's necessarily going to be 50/50 for every single person who does the flipping. I think that our reasoning mind will get fixated on the 50/50 thing, and we will tend to think that a random string of flips cannot exceed a certain number (I chose the number 15 because it seemed reasonable to me) of same results in a row, when in fact the number of consecutive heads or tails results could be very long, and far more than 15.
What I am saying is that the people who did the fake work are much more likely to have smaller ';runs'; of the same flip in their made up list. There will be some limit (and I don't know what that limit is--for me it appears to be 15) to what they can imagine as a ';long string'; of the same consecutive flips. The reality is that if you are doing 200 flips, you may have a string of the same flip which is so long that no one would ever imagine it to be possible, and thus would never include it in a made up record of flips.
Does that make any sense?
I think the same thing happens when we gamble. I have heard people say, ';I must surely hit the jackpot on this slot machine soon, because I have pulled the lever almost 100 times, and the casino advertises a 98% payout rate on their slots.'; Yes, that advertised payout rate does mean that at some point the machine is going to pay out 98% of what was paid into it, but it doesn't mean that it's going to happen once you pull the lever the 100th time. Our brains just automatically think 98% and 100% are related in a very linear fashion, when they are not. The reality is that that 98% payout can take days and days to be realized. We don't know where exactly 100% lies. Our brain tells us that 100% correlates to 100 pulls of the lever, but that is a fallacy. For all we know, that lever needs to be pulled 673 times for the machine to decide that is 100%, and finally hit a jackpot which pays back 98%.
And therein lies the trap of gambling. We live in a base 10 world, and we don't really understand the nature of random acts. We tend to think of things in increments of 5 or 10. When someone plays poker, they tend to think, ';If I play 100 hands, surely I will win at least 5% of the time, or 5 hands.'; What they don't realize is that because that deck is shuffled before each hand, they have exactly the same odds of winning on every single hand. There is no cumulative effect. I don't know enough about gambling to know off hand what the odds are of a winning hand, but I know that there are all sorts of variables, including what kind of poker you are playing and how many people are at the table. Hell, even someone cutting the deck makes a difference, as each person cuts a deck slightly differently (each person shuffles differently, as well, in a game where manual shuffling takes place, and there is no house dealer, only player dealers). We try to bring order to gambling because we imagine that there must surely be a limit to how random anything in the universe can be. And that's where we fool ourselves. Things can be more random than we imagine.
So there. I went ahead and went out on a limb and made a fool of myself. Did I even manage to address the question? Or where you asking about the best detergent for an automatic dishwasher???Coin tossing (True nature of random outcomes)?
Over the 200 tosses. it can be expected that there will be reasonably close to half heads and tails. The professor is looking for those papers that are close to this.
Actually, the side of the coin with the head on is ever so slightly heavier than the other side, and so all though it will be close, it will land tails up more times than heads.
I think i've heard of this one before. if i recall correctly, the ones that were faked had exactly 100 heads and 100 tails.
In the world of random events there is something called variance that is SOMEHWAT predictable, though anythign can happen, if you play enough trials you will find that most of the time the results will fall within a bracket, as you add more trials the results will aproach and get closer to the mathematical average and fall within a smaller and smaller bracket a very high percentage of the time. Using this as a guide he could isolate papers outide thr bracket specified by a certain lowe probibility of variation and claim that it is very likely papers outside the bracket were faked.
He could never for sure claim that a paper was fake, however, he could be 90% accurate (or some number around there) on claiming papers are fake which may be good enough for grading purposes.
Anything could happen so he could never be 100% sure or 100% accurate, he could approach 100% though.
Another tactic, as suggested above by other posters is that if a paper had exactly a 50% head to tail ratio, it is likely that it was faked since the probibility of landing EXACTLY on the average with that number of trials is very low, howver it does not mean ti will not happen.
Regarldess he will never know for sure, only be able to speculate with a high degree of accuracy, im sure if you broguht up that point when he failed you for your paper, he would immediately pass it since that is probibly what he is tryign to teach anyways.
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