Flippin coins
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Ok here's something which I've debated with friends and co-workers in the past. Now it is common knowledge that if you flip a (perfectly balanced) coin the outcome will not be influenced by the outcome of previous flips, there will always be a 50/50 chance that head will come up and equally so for tails, I completely agree with this. However let's say hypothetically we flip a coin 1000 times, then in a perfect enviroment 500 should be heads and 500 should be tails. Ok then let's say on the first 500 flips even with a perfect coin in a perfect enviroment that heads came up every single time (that will probably lead to the conclusion the coin is not balanced, but lets just say it is)... then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? As I mentioned I totally agree with notion of each flips being completely independent of any previous outcomes, but yet everytime I ponder this notion I can't help bringing in the latter scenario which on my reasoning seem to contradict it. Anyone wanna comment or explain to me where my reasoning is faulted?
in a perfect environment, the 500:500 outcome is unlikely. However, it is the most likely one, and if you repeat this test will see most outcomes around 500:500 +/- 150. In "human sense" reasoning, there is (I think) a common flaw in causality. There is no mechanism adjusting the outcome, or "pushing" it towards 50/50. Rather, all that statistics says is you shouldn't bet your life on 500-in-a-row. And it's suprisingly good at it.
Pandoras Gift #44: Hope. The one that keeps you on suffering.
aber.. "Wie gesagt, der Scheiss is' Therapie"
boost your code || Fold With Us! || sighist | doxygen -
Ok here's something which I've debated with friends and co-workers in the past. Now it is common knowledge that if you flip a (perfectly balanced) coin the outcome will not be influenced by the outcome of previous flips, there will always be a 50/50 chance that head will come up and equally so for tails, I completely agree with this. However let's say hypothetically we flip a coin 1000 times, then in a perfect enviroment 500 should be heads and 500 should be tails. Ok then let's say on the first 500 flips even with a perfect coin in a perfect enviroment that heads came up every single time (that will probably lead to the conclusion the coin is not balanced, but lets just say it is)... then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? As I mentioned I totally agree with notion of each flips being completely independent of any previous outcomes, but yet everytime I ponder this notion I can't help bringing in the latter scenario which on my reasoning seem to contradict it. Anyone wanna comment or explain to me where my reasoning is faulted?
I don't see how this would be possible. It takes a long time to flip a coin 500 times, let alone 1000. If it were me doing the flipping, there is a 100% probability that I'd have already spent it before completing the 1000 flips, probably on some chicken nuggets or something. The generation of random numbers is too important to be left to chance.
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yaddaYoda wrote: Anyone wanna comment or explain to me where my reasoning is faulted? First of all, the probability of obtaining 500 heads in 500 flips is extremely small, especially for a binomial process. Because the flipping process is non-Markovian, future flips will occur independently of the past (and current) history of outcomes. That is, the previous 500 flips and the current flip result will not influence the outcome of the next, or even next 500, flips. Thus, in actuality, the probability of obtaining 500 tails in a row on subsequent 500 tosses is also minutely small. However, the chance of obtaining a head on toss 501 will still be 50/50 for a (non-Markovian) binomial process. Your reasoning is wrong because you are claiming that the previous 500 events will influence the next event or possibly next 500 events. This is not true, particularly for a non-Markovian process. I think you need to differentiate between an outcome and a sequence of outcomes. It is the sequence of outcomes in this case that has a small probability of occuring. I think you need to imagine a small sample space first. Let's consider three four flips. Thus you have the following combinations: HHHH HHHT TTHH HTHT THHH HTTT HTTH THHT HTHH THTT HHTT TTTH HHTH TTHT THTH TTTT Obviously events HHHH and TTTT have a smaller probability of occuring than do the other events. This is because these events belong to the tail of the distribution and are less likely to occur. It is not that any individual outcome in the event influences the probability of the next outcome. It is just that the event itself is improbable. Thus, if I flip two heads on the first two trials, it is more likely that eventually an event other than HHHH occurs, even though the outcome of the flip is 50/50. Extending this reasoning to your 500 heads example we can see that as an event, 501 heads in a row is less likely than 500 heads and then a tail on the 501 flip. Additionally, we could also appeal to the law of large numbers and central limit theorems to demonstrate that this cannot occur. I will leave that to you. This[^] site may prove interesting in that respect. John Theal Physicist at Large Got CAD? http://www.presenter3d.com[
Another cogent & sage response, John - I was going to respond with something to do with independence, but you beat me to it - and with a lot more information :-) Stuart Dootson 'Java, Basic, who cares - it's all a bunch of tree-hugging hippy cr*p'
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Ok here's something which I've debated with friends and co-workers in the past. Now it is common knowledge that if you flip a (perfectly balanced) coin the outcome will not be influenced by the outcome of previous flips, there will always be a 50/50 chance that head will come up and equally so for tails, I completely agree with this. However let's say hypothetically we flip a coin 1000 times, then in a perfect enviroment 500 should be heads and 500 should be tails. Ok then let's say on the first 500 flips even with a perfect coin in a perfect enviroment that heads came up every single time (that will probably lead to the conclusion the coin is not balanced, but lets just say it is)... then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? As I mentioned I totally agree with notion of each flips being completely independent of any previous outcomes, but yet everytime I ponder this notion I can't help bringing in the latter scenario which on my reasoning seem to contradict it. Anyone wanna comment or explain to me where my reasoning is faulted?
Like John mentioned, the throws are independent so past events have no influence on future events. Given that and the fact that the coin is 50/50 for each throw, you can work out the probability of each outcome. For example after 3 throws: 3H 0T = 1/8 2H 1T = 3/8 1H 2T = 3/8 0H 1T = 1/8 You can carry this out until you hit 500 and you will find the chance of 500 heads to be 2-500, and then the chance of getting 500 tails is again 2-500 which is just as unlikely as getting the first 500 heads. One of the first complications is the relationship between the grouping buckets like 500H 500T and the entire ordered sequence of events. According to bucket recordkeeping, 500 heads followed by 500 tails is the same as alternating heads and tails. However things get confusing for the 500 heads case because the bucket only contains 1 possible sequence of events and people tend to confuse the bucket and the sequence. Another complication is real life. Since 500 heads is so unlikely for a fair coin, our real life experience tells us that it is much more likely that we have a biased coin or process. This is really hard to get over. In fact, if there is even a one in a billion chance that you have a coin that only throws heads, it is much much more likely that you have a biased coin then you got 500 heads with an unbiased coin. Yet another difficult thing to wrap your head around is the relationship between sequences and individual trials. Since we are dealing with events with 2 outcomes we can view the sequences as chains made up of orange and green links. Now in this view, the event independence is a statement about the relationship of a link given it's closest neighbors. What you are doing is taking that rule about links and extending it to discover relationships between long chains. It isn't at all obvious how those rules relate to each other.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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yaddaYoda wrote: Anyone wanna comment or explain to me where my reasoning is faulted? First of all, the probability of obtaining 500 heads in 500 flips is extremely small, especially for a binomial process. Because the flipping process is non-Markovian, future flips will occur independently of the past (and current) history of outcomes. That is, the previous 500 flips and the current flip result will not influence the outcome of the next, or even next 500, flips. Thus, in actuality, the probability of obtaining 500 tails in a row on subsequent 500 tosses is also minutely small. However, the chance of obtaining a head on toss 501 will still be 50/50 for a (non-Markovian) binomial process. Your reasoning is wrong because you are claiming that the previous 500 events will influence the next event or possibly next 500 events. This is not true, particularly for a non-Markovian process. I think you need to differentiate between an outcome and a sequence of outcomes. It is the sequence of outcomes in this case that has a small probability of occuring. I think you need to imagine a small sample space first. Let's consider three four flips. Thus you have the following combinations: HHHH HHHT TTHH HTHT THHH HTTT HTTH THHT HTHH THTT HHTT TTTH HHTH TTHT THTH TTTT Obviously events HHHH and TTTT have a smaller probability of occuring than do the other events. This is because these events belong to the tail of the distribution and are less likely to occur. It is not that any individual outcome in the event influences the probability of the next outcome. It is just that the event itself is improbable. Thus, if I flip two heads on the first two trials, it is more likely that eventually an event other than HHHH occurs, even though the outcome of the flip is 50/50. Extending this reasoning to your 500 heads example we can see that as an event, 501 heads in a row is less likely than 500 heads and then a tail on the 501 flip. Additionally, we could also appeal to the law of large numbers and central limit theorems to demonstrate that this cannot occur. I will leave that to you. This[^] site may prove interesting in that respect. John Theal Physicist at Large Got CAD? http://www.presenter3d.com[
This makes sense mathematically, but what usually ends up happening more often in real life is that there is a very small chance that you have a biased device and that is most likely the cause of results like this. So common sense and a larger statistical model tells us that we probably have a biased coin. That experience is one of the hardest things to get over when dealing with statistics, or at least it was for me.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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Ok here's something which I've debated with friends and co-workers in the past. Now it is common knowledge that if you flip a (perfectly balanced) coin the outcome will not be influenced by the outcome of previous flips, there will always be a 50/50 chance that head will come up and equally so for tails, I completely agree with this. However let's say hypothetically we flip a coin 1000 times, then in a perfect enviroment 500 should be heads and 500 should be tails. Ok then let's say on the first 500 flips even with a perfect coin in a perfect enviroment that heads came up every single time (that will probably lead to the conclusion the coin is not balanced, but lets just say it is)... then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? As I mentioned I totally agree with notion of each flips being completely independent of any previous outcomes, but yet everytime I ponder this notion I can't help bringing in the latter scenario which on my reasoning seem to contradict it. Anyone wanna comment or explain to me where my reasoning is faulted?
That kind of thinking is why people waste money on lottery tickets and roulette wheels. Because my number hasn't hit for so long, it *just has to* hit this time. There is always a 50% chance of flipping heads or tails. It doesn't matter what the previous flip was, or what the previous million flips were. Now if you flip the coin and it lands on the edge, that's a statistically relevant event! Jeff Martin My Blog
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This makes sense mathematically, but what usually ends up happening more often in real life is that there is a very small chance that you have a biased device and that is most likely the cause of results like this. So common sense and a larger statistical model tells us that we probably have a biased coin. That experience is one of the hardest things to get over when dealing with statistics, or at least it was for me.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
andy brummer wrote: So common sense and a larger statistical model tells us that we probably have a biased coin. Yes. This is very true. He had assumed a fair coin, so I treated it as such. Your post below treats this very well. John Theal Physicist at Large Got CAD? http://www.presenter3d.com[^]
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Ok here's something which I've debated with friends and co-workers in the past. Now it is common knowledge that if you flip a (perfectly balanced) coin the outcome will not be influenced by the outcome of previous flips, there will always be a 50/50 chance that head will come up and equally so for tails, I completely agree with this. However let's say hypothetically we flip a coin 1000 times, then in a perfect enviroment 500 should be heads and 500 should be tails. Ok then let's say on the first 500 flips even with a perfect coin in a perfect enviroment that heads came up every single time (that will probably lead to the conclusion the coin is not balanced, but lets just say it is)... then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? As I mentioned I totally agree with notion of each flips being completely independent of any previous outcomes, but yet everytime I ponder this notion I can't help bringing in the latter scenario which on my reasoning seem to contradict it. Anyone wanna comment or explain to me where my reasoning is faulted?
yaddaYoda wrote: then statiscally shouldn't the next 500 flips be all tails and hence making the chance of tails coming on flip 501 be higher than 50/50? No, this is commonly referred to as the "law of averages" even though it's pure fallacy. Besides, why is 500H followed by 500T "perfect"? Why not 250H-250T? Or alternating H and T? --Mike-- LINKS~! Ericahist | 1ClickPicGrabber | CP SearchBar v2.0.2 | C++ Forum FAQ Strange things are afoot at the U+004B U+20DD
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in a perfect environment, the 500:500 outcome is unlikely. However, it is the most likely one, and if you repeat this test will see most outcomes around 500:500 +/- 150. In "human sense" reasoning, there is (I think) a common flaw in causality. There is no mechanism adjusting the outcome, or "pushing" it towards 50/50. Rather, all that statistics says is you shouldn't bet your life on 500-in-a-row. And it's suprisingly good at it.
Pandoras Gift #44: Hope. The one that keeps you on suffering.
aber.. "Wie gesagt, der Scheiss is' Therapie"
boost your code || Fold With Us! || sighist | doxygenpeterchen wrote: causality A vision of Monica Bellucci always flashes past my minds-eye when I read or hear that word these days. Damn you Matrix! regards, Paul Watson South Africa The Code Project Gary Wheeler wrote: It's people like you that keep me heading for my big debut on CNN...
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peterchen wrote: causality A vision of Monica Bellucci always flashes past my minds-eye when I read or hear that word these days. Damn you Matrix! regards, Paul Watson South Africa The Code Project Gary Wheeler wrote: It's people like you that keep me heading for my big debut on CNN...
Paul Watson wrote: Damn you Matrix! I feel for you, brother. really. Monica Bellucci ..pfft.. X| ;) BW
I want pancakes! God, do you people understand every language except English?
Yo quiero pancakes. Donnez moi pancakes. Click click, bloody click pancakes!
-- Stewie Griffin -
peterchen wrote: causality A vision of Monica Bellucci always flashes past my minds-eye when I read or hear that word these days. Damn you Matrix! regards, Paul Watson South Africa The Code Project Gary Wheeler wrote: It's people like you that keep me heading for my big debut on CNN...
Truth be told, there are worse things to flash by :rolleyes:
Pandoras Gift #44: Hope. The one that keeps you on suffering.
aber.. "Wie gesagt, der Scheiss is' Therapie"
boost your code || Fold With Us! || sighist | doxygen