Coin flip probabilities and relevance

[Hanshananigan]

I saw this thread at the top of a Google search, so I thought it might be worth an illustration of why the OP's logic doesn't hold up.

I used Excel to create 104 sets of 400 "coin flips." By the OP's logic, if the first 200 flips in each set tended to be more heads or more tails, the second 200 flips should have "pressure" to reverse that trend to result in a closer to 50/50 split. Ignoring the few sets that had 100 "heads" in the first 200 flips (50/50 split, so there would be no "pressure") and those that had 100 "heads" in the second 200 flips, I counted how many sets had >100 heads in the first 200 flips and <100 heads in the second 200 flips and did the same thing for tails (perhaps providing evidence of "pressure"), and calculated a rate of how many trials supported the "pressure" hypothesis (ignoring those trials with 100 "heads" in the first or second 200 flips). To keep things simple, I ran each set of trials 10 times and calculated the rate for each set.

It was expected that the rate of trials supporting the "pressure" hypothesis would be significantly > 50%.

The results: 56, 49, 53, 53, 48, 39, 45, 48, 52, 37

That is, an average 48% of the time, if #heads > #tails in the first 200 flips, that trend would be reversed in the second 200 flips. Thus, no evidence of "pressure."

In terms of application on the roulette table, if you have enough cash for 40 spins, and you know the outcomes of the first 20 tended to be "black," you will not necessarily come out ahead if you bet on "red" for the next 20 spins.

Looking at it another way, for those trials in which at least 110 flips of the first 200 were heads, the 201st flip was tails only an average of 45.1% (36, 40, 56, 60, 64, 44, 58, 67, 50, 40). Although 45% seems like a decent spread for betting, the variance across trials is huge due to the infrequency of finding at least 110 flips among 400 trials, so the confidence interval will be wide. Again, even if the roulette table has tended to run "black," it does not affect subsequent spins. Not worth betting on.

These outcomes will be expected by most of those who posted on this thread. For anyone else, I hope this example provides a link between the OP's logic and the properties of probability offered by others.

 

[heyhay4062]

As others have not so politely stated get a clue. Just because there is a statistical anomaly there is no”probability force” that will make it correct itself. It may never correct itself, or more correctly stated it may take an infinate number of chances to correct itself. At every odd numbered toss you are guaranteed a statistical anomaly. After one toss it will be either heads or tails. You are literally saying that if you toss a coin twice if the first result is heads, then the second toss is bound to be tails. Which hopefully you understand isn’t the case...

 

[roryjester]

I'm not a mathmatician or physicist, though I have a couple of degrees in the sciences, the specifics of which I will leave out in case I will be belittled for irrelevance here. I did take one probability and statistics course at university a long time ago, an introductory one which I somehow passed and actually did ok at, though I came out not really getting 'it'. Like how much of it really can be applied for any 'specific' instance in life, being based usually on things like infinity or at least, a large enough sample size or population, as well as all kinds of manipulations of figurative mind and pencil, and predictions good only on a large numerical scale (although good for big entities like corporations predicting profits and armies predicting all sorts of things that are necessary for military success, et cetera). Being a gambler at heart and actuality, I've experienced more than I've ever thought, especially about things like flipping a coin, which may be a serious disadvantage in some people's thinking, but not so much in the minds of others who based most of their actions and decisions on the unpredictability of life and real experience. I've read thru the entire thread and though I don't understand a great deal of what's being used to argue each poster's particular points, I do find it interesting, at least from my narrow limited point of view and comprehension. Let's begin with my two bits worth. Since infinity is something almost impossible to grasp, except maybe abstractly, like mathmatical singularities, it might not be such a great idea to use it to argue more mundane things like the flipping of a coin. If I comprehend right, in an infinite sample size which of course means the inclusion of all flips or sets of flips ever performed or imagined in the universe from the infinite past (debatable) to the infinite future (again debatable), the number of heads and tails will (or have already) come out to a 1:1 ratio or 50/50. Just true randomness or unpredictabiltiy at work to give us a final predictable or non-random number, I guess. Now in any 'finite' sample of flips, anything can happen including a million flips in a row which result in all heads, let's say. This would be a great statistical anomaly, but funnier things than this have happened, like perhaps the greater improbability of human life (or any kind of 'life') actually starting up in this great big universe (yet that's what has happened). Now, the OP's question as I understand it, is: Is there pressure for the next million or so spins after the 'all-heads' sample to favour more tails than heads? I'd say for any individual flip after that, the probability would be 50/50 just as if the coin never knew it flipped heads a million times in a row before that (heads again, baby!?). But the OP's question is really bigger than that. He's saying in the infinite minus 1 millions spins after that, is there going to be more tails than heads? This is just my gambler's intuition speaking, but I would say yes, although not by much (the ratio will still be approaching 50:50 for all 'practical' matter). So if u got infinite amounts of money, time, and patience, it might not be a bad idea to bet on tails in the infinite time after u see the first million flips go all heads, although another question to be asked is: Were u there to see the previous million flips before the all-heads streak, cos u know, it might have been a 'million-all-tails' result set before that; then ur back where u started: 50/50 and no real or perceived 'pressure' to compensate for older statistical anomalies. Please inform me if my jerry-rigged gambler's intuition is wrong here somehow. By the way, does anyone here know if slot machines are truly random, or do I just have to stay with a cold machine until it is 'pressured' into becoming hot again (so I can get all my money back)? I understand however that the 'payout-percentage' programming (involving a 'truly' random number generator?) may not be based on infinite 'spins' though (maybe a million, a billion, even a gazillion, but not infinite). By universal law, it has to pay back a certain percentage of the finite money put into it in a finite time. It's just predicting those times (or the length of time before payouts) that's the 'infinite' problem, isn't it? Aah, what the hell am I talking about? Cmon, smile, be happy.

 

[CRGreathouse]

Since infinity is something almost impossible to grasp, except maybe abstractly, like mathmatical singularities, it might not be such a great idea to use it to argue more mundane things like the flipping of a coin.

Actually infinite numbers are pretty easy to pick up (depending on the type you pick); I'm not sure where this idea comes from, though it's common. But you're right that we don't need it; there are finitistic ways to thinking about it. Here's one:

For any positive percentage (say, 0.1%) and any certainty less than 1 (say, 99%), there is an M such that for all N > M,
the chance that N fair coin flips will fall between 50% - the percentage and 50% + the percentage (49.9% to 50.1% in this example) is at least the specified certainty.

Now, the OP's question as I understand it, is: Is there pressure for the next million or so spins after the 'all-heads' sample to favour more tails than heads? I'd say for any individual flip after that, the probability would be 50/50 just as if the coin never knew it flipped heads a million times in a row before that (heads again, baby!?). But the OP's question is really bigger than that. He's saying in the infinite minus 1 millions spins after that, is there going to be more tails than heads? This is just my gambler's intuition speaking, but I would say yes, although not by much (the ratio will still be approaching 50:50 for all 'practical' matter).

Those contradict each other! If each following coin flip is unbiased, then the collection of coin flips will also be unbiased.

So if u got infinite amounts of money, time, and patience, it might not be a bad idea to bet on tails in the infinite time after u see the first million flips go all heads, although another question to be asked is: Were u there to see the previous million flips before the all-heads streak, cos u know, it might have been a 'million-all-tails' result set before that; then ur back where u started: 50/50 and no real or perceived 'pressure' to compensate for older statistical anomalies. Please inform me if my jerry-rigged gambler's intuition is wrong here somehow.

Yes, intuition has failed you this time. It happens.

By the way, does anyone here know if slot machines are truly random, or do I just have to stay with a cold machine until it is 'pressured' into becoming hot again (so I can get all my money back)?

Slot machines are generally random, modulo concerns about their use of pseudorandom numbers rather than true RNGs (don't worry about it; it doesn't affect your question). But one slot machine need not be like another. It's possible to have one machine in a room that pays out more often than others in that same room -- and from what I hear, that's not uncommon. So within a machine, it's essentially random, but between machines I wouldn't expect similar long-term results.

 

[praeclarum]

Thanks for the unridiculed (almost) reply.
If there is no pressure to return to 50/50, then why doesn't one just flip heads indifinitely?

The fallacy in your reasoning is in making the assumption that because something is likely to happen, then it naturally tends to that. While this is true to some extent, it is only indirectly - it is a product of the fact that as n (number of tosses) approaches infinity, the heads to tails ratio approaches 1:1, simply because as n increases, it is increasingly improbable for you to keep up a streak of all tails and all heads that just happens to comply with the data.

Here's a mini-demonstration:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
...

This is Pascal's triangle, but it represents the relative probability of getting a certain number of heads out of the tosses. So if you flipped a coin 6 times, there is a 10 / 32 chance of it getting 3 heads, whereas there's only a 1 / 32 chance of it getting 0 heads.

Anyway - a little off-topic, I guess. Big picture - you're mistake is in saying that the relative higher chance is because of some "law of averages" that has a will of its own.

 

[Stephen Tashi]

I think these two qualitative statements need not be contradictory:

1) An imbalance in the first N tosses of a fair coin gives no information that improves our prediction of the result of the next M tosses of it.

2) As the sequence of tosses of a fair coin progresses it is likely that there will be times when the total numbers of heads has a big lead over the total number of tails or vice versa.

As I recall, one of Feller's books discusses 2) in a mathematically rigorous way. One can compute the probability of one result or the other taking a lead of a certain size. It's understandable that people who look at graphs of real or simulated sequences of coin tosses get the impression that swings one way are balanced by swings the other way.

It would be interesting to see the results from a model of coin tosses where the tosses are not independent and a specific formula is given to describe the dependence. For example suppose for toss i < 7 the probability of a head is 1/2 and for i > 6 the probability of a head is given by: 1/2 + (0.4)(3-K)/3 where K is the number of heads in the previous 6 tosses.

 

[yudiski4]

Forget "waves", there are none. Try to look at it this way. In an infinite number of tosses the heads/tail ratio will come out to very, very close to 50%/50%. Agreed? And in that infinite number of tosses, there will have been, almost assuredly, a streak of 1000 straight heads. Also agreed? But the wave theorist says, "Woah, after those 1000 heads, assuming it was running close to 50/50 up to then, there would have to be a "tail wave" for it to end up 50%/50% at the end".

But the fallacy is: in infinity, there is no "end". The intuitive force that makes the "wave" seem inevitable is tied up in the human brain's inability to conceive of or think in terms of infinity.

 

[Parahandy]

We are almost agreed on the very cose to 50:50 on infinite coin tosses (arguably it's exactly 50:50 but who's arguing?). But your 1,000 straight heads is a bit far fetched and that is the fundamental problem of infinity - it has been said (in some BBC programme) that if you attribute a number to all letters of the alphabet then the infinite progression of Pi would hold every book that has ever been written and every book yet to be written. I understand the logic, which interestingly becomes more philosophical than physical and therefore is never going to happen. This is where progressive logical steps end up in a ridiculous proposition and so is the idea of coin tosses producing a 1,000 heads in a row.

I haven't done it but I guess if you plot the incidence of coin tosses starting at 1 head, 2 heads in a row, 3 heads, e.t.c. you will end up with a front end skewed curve with a mean around 4 or 5. You might manage to throw 10 in a row but thereafter, at some point not too far away, your chance runs out to zero. I said it before, the concept of infinity must be flawed; it highilghts a deficiency in our ability to percieve our surroundings, i.e. there is something wrong with maths and numbers and our logic - irrational numbers are just that - irrational.

Coming back to the question, disagreeing with your 1,000 heads makes the point. It is a fact that the coin tosses will pass through the 50:50 and err for a while on the tails side as much as heads. They will switch from one result to the other sooner rather than later and one run can't carry on indefinitely (the laughable thing about infinity is that, if you believe in it, there will be, at some point, an infinite run of heads but, of course, the same would apply to tails - and to the coin landing on its edge).

The problem with probability is that there are some certainties about it but no discernable pattern that we can see. It is as if there is something hidden that is discoverable and would solve the problem. If that is true and someone discovers the solution, it would turn maths on its head and (here's a philosophical point) may destroy reality as we know it because we would have certainty of the future - and that's just not allowed!

The fallacy is the whole concept of infinity.

 

[yudiski4]

"The problem with probability is that there are some certainties about it ... The fallacy is the whole concept of infinity."

If you need to have certainties, and you outright reject the concept of infinity, then the study of probabilities is going to lead you to a stone wall.

 

[chiro]

Forget "waves", there are none. Try to look at it this way. In an infinite number of tosses the heads/tail ratio will come out to very, very close to 50%/50%. Agreed? And in that infinite number of tosses, there will have been, almost assuredly, a streak of 1000 straight heads. Also agreed? But the wave theorist says, "Woah, after those 1000 heads, assuming it was running close to 50/50 up to then, there would have to be a "tail wave" for it to end up 50%/50% at the end".

But the fallacy is: in infinity, there is no "end". The intuitive force that makes the "wave" seem inevitable is tied up in the human brain's inability to conceive of or think in terms of infinity.

You don't need to think of it necessarily in terms of infinity, but rather in terms of something "really large".

For many practical purposes the strong law tells us a lot about the kind of limiting probabilities for large enough sample sizes as it would for an infinitely large number of them.

To understand this its best to think of the derivative of 1/x. If x is big enough then any change thereafter is not going to have much of an effect if the observations up to that point reflect a mostly unbiased sample. If the sample is highly biased then we can't necessarily do this, but for most purposes "large enough" samples will provide a distribution that is good enough to represent the true distribution for "infinite" sample sizes.

 

[TechnocratX]

I know this thread is over 8 years old, but this reply is for the benefit of someone like me who stumbles across it. Plus I think I can explain it in a more simpler manner, especially for those with basic stats knowledge.

Concerning coin flip probabilities...

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.

Any thoughts, suggestions, comments

Ok, say you did the first 10,000 coin flips, and got 9000 heads. This gives you a 90/10 distribution. Now you're thinking you're at the top of a heads wave, and should expect a tail wave to take you back to a 50/50 distribution.

Then you carry on and do another 1,000,000 coin flips, but this time you get exactly 500,000 heads and 500,000 tails. So no increase in tails from a pressure wave. But, even without the tail pressure wave your graph has now moved to a 50.4/49.6 distribution.

What's happened is that you've simply increased the sample size and that has reduced the effect of the 9000 heads. Hopefully you can now see that the wave patterns tending towards the 50/50 distribution, are caused by the increase in samples and not an increase in heads or tails through a pressure wave.