Concerning coin flip probabilities..... In my graduate undergrad & grad stat classes I learned the probability of getting heads or tails is 50/50. But I have something to toss out into the ring for comment.... According to my limited research, the 50/50 probability appears to be a mean probability, and that the actual probability is relevant upon previous coin flips in order to obtain an mean 50/50 probability. For example, if out of 10,000 coin flips, I get 9000 heads, then for the next 10,000 flips, the distribution of heads vs. tails would not be 50/50, but would be weighed in favor of more tails in order to get back to the 50/50 mean. I call such a change in normal tendency as "probability pressure" (PP)on the "probability wave" (PW). I realize the term probability wave is already established in reference to light, but it seems to apply here. If one graphs the results of 10,000 coin tosses (or 100,000 as I have), giving heads a value of +1 and tails a value -1, one can easily visualize the PW and should be able to recognize the strength of the PP, either positive or negative, seems to increase the greater the distance from the mean “score” of 0 the wave extends. Considering the range from the crest of one wave to the next, and the distance between the crests, one may theorize that at the peak of each wave the odds are not truly 50/50, but are skewed. One may also see that there appears to be a limiting factor on the actual height, or frequency, of the wave, as the possible range for 10,000 tosses could theoretically be a score of 10,000 (100%) either positive or negative, but I have not observed a variance of more than about 3%. A question I have yet to solve is developing a formula to determine the true probability of a coin toss when relevancy is considered. It appears that the higher, or lower, the score from the mean probability, the greater the skew from 50/50, perhaps on some type of ratio. Any thoughts, suggestions, comments
Please let me know what school you attended so that I can avoid hiring any of its graduates. Thanks, Njorl
Njorl I know what I am presenting is "out of the box", but please patronize me and answer these two questions: 1.Have you ever graphed several thousand random numbers in the manner I described, and 2. If you have, have you observed something different?
The chance of rain is 50% today. Is that 50% for a particular point, region or averaged space; point, region or averaged time; particular person; mathematical abstraction; collapsed or uncollapsed wavefunction; butterfly in Japan...?
I don't know about butterflies, but attached here are the results of a probability study I conducted within the past 5 minutes graphed into the wave. I guess what this means is if I walk into a casino with a true roulette table with no 0's (now that's a serious non-probability)and I am told that for the past 40 hours nothing but reds have come up, then a black comes up, that I shouldn't make a side wager that in the next 40 hours more blacks should show than reds, especially since I have never documented more than 35 hours of only one color appearing? Answer after you look at the graph.
You absolutely should not. This is just evidence you have no idea what probability means. As a result, Njorl requested that you tell us where you've taken your graduate probability classes, so that we can all be sure to never take anyone from that school seriously. - Warren
I forgot to add that in the probability analysis I previously graphed, 2497 times I obtained a heads, 2503 times I received a tails. If you look at the graph you will see a low score of -56 approx 400 throws from the last 0 score, a return to 0 in approx 550 throws (an approx 60/40 probability from the low point), then a peak score of approx 32 after 300 more throws, returning to 0 again after about 140 throws ( an approx 70/30 probability from the high point). The range from -56 to 32 (88) is covered in approx 850 throws (a approx 60/40 probability). The wave continues until we reach the end of the graph with an almost 50/50 distribution. You will see similair results no matter how many times you run the analysis, and no matter where you start your count from as long as you have a high number of throws (random numbers) So the overall odds are 50/50, but again, those odds are relevant upon where you are on the wave. I was hoping some one out there who has attended a school you might want to hire from may instead of ridculing my objective observation, may instead attempt to explain and expound upon it. Such is the role of the greatest minds, while lesser ones just regurgitate what they've been fed. Again, does someone have an explanation instead of put downs and ridicule as if from the Flat Earth Society?
What you're graphing is called a "one-dimensional random walk." You'll only obtain 50% heads and 50% tails in the limit as your sample size approaches infinity. This is the definition of probability. Wrong. The probability is exactly 50%, all the time, as defined above. If you had an infinite sample, you'd have exactly 50% heads and 50% tails. This does not mean that any finite sample will have 50% heads and 50% tails, however. Naturally, as your sample size decreases down to one sample, it obviously is either 100% heads or 100% tails. Your misconceptions are at a middle-school child's level. I highly doubt that you've ever studied statistics at the collegiate level if you can't even grasp the very definition of probability. - Warren
Thanks for the unridiculed (almost) reply. I did achieve a 50/50 result for all intents and purposes in my 5k example, and in my own 100k example Of course the larger the number of tosses, the closer to the results would come to an infinite number of tosses...10,000 is better than 10. So are you saying that there is no probability wave, that even if infinite, there are no swings in the results of a coin toss, that once some one gets let's say 200 heads ahead that if he keeps tossing indefinitely he would never get 200 tails behind then back to 200 heads ahead? And if one is 200 heads behind he would never even get back to being even? If there is no pressure to return to 50/50, then why doesn't one just flip heads indifinitely? I am not trying to overrule probability, just looking at it from a different perspective.
No. I never said anything about any finite samples. I said, very specifically, that you'll get 50% heads and 50% tails ONLY in an infinite sample. If you had a sample of 1000 flips, it is entirely possible that all 1000 are heads. This would be a large departure from what you may intuitively feel is "random," but it's not. Every possible sequence of 1000 flips is equally likely. It just happens there is only one such sequence with 1000 heads in a row, while there are very many sequences with roughly the same number of heads and tails. In other words, for ten flips: TTTTTTTTTT and THHTTHTHHT are exactly equally likely to occur. There is no "pressure" to return to 50/50. As I've said, you could have any finite sample size, as big as you like, and fill it with tails. This is exactly as likely as any other combination. There's nothing special about it. There is no "pressure" for finite samples, only in the limit as your sample size approaches infinity. This is a common fallacy, and the reason why gambling is so addictive: you go into a casino and see that a roulette wheel has spun red 10 times in a row -- and somewhere, deep inside, you think "the next one just HAS to be black!" But no, it doesn't. It could spin red for three hundred million years straight, just as easily as it could spin anything else. Your intuition that this is improbable is simply a result of the fact that there are many more sequences of roughly half red and half black than there are sequences of all red or all black. The very definition of a random process is that each experiment is independent of history. In the case of a coin being flipped, there is absolutely no dependence on its history, and thus, no pressure of any sort to return to 50/50. Richard Feynman summed up this fallacy by exclaiming to a class that he had just seen a license plate in the parking lot that read ARW 457. "Imagine that!" he exclaimed, "of all the millions of license plates in the state of California, imagine how amazingly, mind-bogglingly improbable it was for me to see that one!" Right, the perspective of a middle-schooler. - Warren
The "pressure" mentioned is more the result of intuitive physical experience than probabilistic-mathematical objectivity. Such perceived pressure derives from anticipating the same random result as that underlying statistical mechanics.
Versace, do you not understand the concept of independent probabilty? The result of each coin flip is independent of the results in the other. It's not quite agreed among mathematicans that an infinite number of trials in a fair coin flip will result in 50% tails 50% heads as some argue that surely each permutation is equally likely and by excluding a particular permutation you exclude all permutations, personally I think there reasoning in this merely shows up the fallacy of considering an infinite number of trials.
Yes I fully understand the prescribed concepts of independent probability, I read the books, aced the tests, and moved on. One of the things my instructor drilled into us was "Look at the data", that true scientific research is based on data, not postulations. But what started me on this particular quest was based upon simple reasoning...... If the rules of probability assume 1. The odds of a coin flip are 50/50 2. The more coin flips, the closer you get to a result of 50/50 Then it stands to reason that if after begining with any coin toss, that after X number of coin tosses, the result is 0/100, then after an indeterminite number of coin flips the result would return back to 50/50 at some point, and probably head towards 100/0. It is this pull back to center that impies the coin flips after 1/100 is actually skewed from 50/50. This asumption was contrary to what I had been taught and is generally accepted until one day I was looking at a bell curve and realized that if one is running truely random samples and the intial results are 5 deviations from the mean, then you should in future trails begin to obtain results that will eventually result in a true bell curve. Now, granted one may not return from 100/0 to 50/50 in a limitied number of trials, but with enough trials, eventually one would. Otherwise, the 50/50 proposition is absolutely false. Why so many gamblers lose their shirts is not because eventually the roulette wheel will show more reds than blacks if only blacks have been appeared for 30 minutes, but the gamblers just run out of money before it does. That's when my probability wave theory developed. So I ran several tests using random numbers in various sizes up to 100,000, and when I graphed the results, sure enough a wave pattern appeared each time, and I would end up with a roughly 50/50 distribution. So if a bell curve is generally accepted in probability, then a probability wave has to exist, as it truly is nothing more than a different way of looking at a dynamic, not static, bell curve, and if a probability wave exists, then the theory of independent probability is unsurped by relevent probability, at least that's what my data tends to imply. Just as light does not naturally travel in a straight line as long assumed, but it's course is relevant, such is true it appears with probability. Again, look at the graph above, run your own numbers, let me know your results.
Wow! All without understanding what you were being taught! No, it stands to reason that if, after X number of coin tosses, the result is 0/100 (you mean 0% one heads, 100% tails for example), it is not necessary that it ever get back to 50%/50%. That is not true and I hope it is not what you were taught. Saying that the chance of heads on a coin flip is "50/50" (or 50%) means that on ANY flip, the chances of heads or tails are equal- "the coin has no memory". What happened in the past has absolutely no affect on what will happen. What is true (the "law of large numbers"- a more precise statement would be the "Central Limit Theorem") is that percentage will tend toward "50/50" as the number of trials gets larger. Suppose you flipped a coin 100 times and it came up heads every time: 100/0. Flip it another 100 times and suppose it still favors heads slightly: 60 heads, 40 tails. You flipped a total of 200 times and get 60+ 100= 160 heads (160/200= 80%) and 0+ 40 tails (40/200= 20%). Now you have 80/20. See? Moving toward 50/50 even though the coin still "favors" heads!
"What is true (the "law of large numbers"- a more precise statement would be the "Central Limit Theorem") is that percentage will tend toward "50/50" as the number of trials gets larger." Then what exactly do you mean by "tend" if not that the end result of more coin flips moves more towards an overall 50/50. "Moving toward 50/50 even though the coin still "favors" heads!" This "moving" is what I describe as pressure to return to 50/50 on the wave if it is "true" there is "tend"ancy to "move" towards an overall 50/50. So by your statements, you agree with me. Again, look at the graph and you will see the wave, run the numbers giving head a value of +1, tails -1, and you will see we both agree.
I'm sorry versace, but I'm not that credulous I simply do not believe that you have ever studied probabilty at any formal level, understanding independent proabiblty is absolutely basic in statistics. There is no 'pressure' other than the law of averages.
JSCD..... so did you even look at the graph? It appears most of you didn't. Again, look at the data, tell me your interpretation. And again, I understand this postulation is contrary to established probability theory, but again, look at the data on the graph above, or run your own, a "probability wave" does exist as Halsoivy stated.. "What is TRUE (the "law of large numbers"- a more precise statement would be the "Central Limit Theorem") is that percentage will TEND toward "50/50" as the number of trials gets larger." If a percentage is "tend"ing, then it CANNOT have an independent probabilty of 50/50, but the probabilty must "tend" toward 50/50 LOOK AT THE GRAPH!!! What you will see is that along the way to tending towards 50/50 you will see wave scores similiar to: -1,-2,-3,-4,-5,-6,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,3,4,5,6,5,4,3,2,1,0,1,0,-1,-2,-3,-4,-3,-2,-1,0...
I despair, in your 'idea' it should either be impossible or there should be less probailty than the expected and observed probailty of getting a sequence of all heads.
You still haven't look at the graph, have you? A (long) sequence of all heads or tails is indeed possible, indeed probable, but then eventually the number of tails would outnumber heads eventually so that somewhere down the continuim of infinity a 50/50 balance MUST occur if the true probabilty is 50/50, regardless of where on that continuim you start. Is that not correct? As HalsofIvy pointed out in his? response, although future tosses may "favor" one outcome, eventually, you must return to 50/50. So if at the beginning of your sequence on the infinite continuim you get 10,000 consecutive heads, sooner or later you will get 10,000 more tails than heads to get back to 50/50 overall, for a string of consecutive heads cannot continue indefinitely.