Impossible coin sequences?

[Alan]

Is a sequence of 100 heads in a row literally impossible to get without cheating?

This was discussed in another thread but a mutual decision has been made to start a new thread about it.
http://www.internationalskeptics.com/forums/showthread.php?t=200394

My position is that it is entirely possible to get that sequence without cheating.

Each flip is approximately 50/50, regardless of what came before. All heads is as likely as any other single sequence.

 

[sol invictus]

A sequence of posts I'm hoping Piggy will respond to:

suppose we play the following game. I make a table with 100 entries. Each entry is either F or NF. Let's say there are 50 Fs and 50 NFs in some more or less random order. Now, here's the game - I flip a coin 100 times. Before each flip I consult the corresponding entry in my table. If it says F, I flip the coin over after catching it, before uncovering it and reading it. If it says NF, I don't flip it before uncovering it.

Do you stil believe that a sequence of 100 heads is impossible, given that setup? Note that if 100 heads is impossible, then whatever sequence you'd get by starting with 100 heads and turning all the Fs into tails is also impossible, since that's what I would have gotten had I not flipped the Fs. But since my table of Fs and NFs was arbitrary, that means all sequences are impossible, which is obvious nonsense.

Therefore, 100 head sequences are possible.

Try that with the example of the cup and stairs and you should see the error you're making.

I read your example, but no, I don't see the "error I'm making". Can you tell me what it is?

While you're at it, can you answer the question I asked you? Is a sequence of 100 heads still impossible if I flip/don't flip the coin after catching it according to a pre-determined plan?

 

[Fnord]

Is a sequence of 100 heads in a row literally impossible to get without cheating?

<snip>

My position is that it is entirely possible to get that sequence without cheating.

Each flip is approximately 50/50, regardless of what came before. All heads is as likely as any other single sequence.

Agreed.

To believe otherwise is to fall for the Gambler's Fallacy; "... the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future."^WP

 

[Safe-Keeper]

Given that you have, for example, 10 coin tosses, and the results are either H or T, the sequence HHHHHHHHHH is no more unlikely than any other, such as HTHHTHHTTH. It is us humans who attribute certain results a special meaning, even though statistically they're no less likely to happen. So yes, it's perfectly possible to get a sequence of a hundred heads. It's just as possible as it is to get any other result, in fact.

 

[AdMan]

Is a sequence of 100 heads in a row literally impossible to get without cheating?

This was discussed in another thread but a mutual decision has been made to start a new thread about it.
http://www.internationalskeptics.com/forums/showthread.php?t=200394

My position is that it is entirely possible to get that sequence without cheating.

Each flip is approximately 50/50, regardless of what came before. All heads is as likely as any other single sequence.

That sequence is just as possible as any other unique sequence of 100 coin flips.

And yes, each flip is 50/50, regardless of what came before.

 

[Vorpal]

The first quote in post #2 really clinches the matter in an intuitive and obvious fashion: all sequences are possible because we can relabel them in an arbitrarily predetermined way, and there is no reason for the universe to care about our labels. I don't understand Piggy's reasons for rejecting this.

---
Throughout that thread, I couldn't make sense of Piggy's argument. Perhaps it's because I live in mathland, but at times he seems to invade that land as well:

It's hardly bizarre to note that long strings of coin flips generate a rugged result space, and that smooth result spaces at great extension are typical of rigged setups.

Do you have any evidence that the statistical model proposed in this case is effective all the way out to the edges, where the long stretches of heads/tails live?

This is suggestive of some sort of "space of sequences", with the special sequences of runs "live" on the edge. So presumably they're special because they're... hard to reach, being on or near the edge?

But for any sequence, there's exactly one possible way to obtain it: the coin has to sequentially go through the exact prescription of the sequence. So they're all "on the edge". If one insists on thinking of the space in terms of reachability, then it looks like a tree:

      *      * = flip
   T/   \H
   *     *   <- 1-sequences: (T),(H)
  / \   / \
 *   * *   * <- 2-sequences: (TT),(TH),(HT),(HH)
    etc.

With exactly one way to reach every possible sequence, and the "edge" being just the final level under consideration. The only way to change that is to introduce some sort of absolute rule that prunes that tree, e.g., "if you have 99 heads, you can only get a tail." So what's the reason for the universe to suddenly start caring? I think that was one of Sol's arguments as well; I just re-motivated it in terms of some space-with-edges that Piggy seems to have a conceptualization of.

 

[Illustronic]

It's also entirely possible to bowl a 300 game, though very much less likely than any other bowling score.

 

[Alan]

It is a very clever argument but I'd say it's a bit difficult to understand.

I love the part that states that if one of them is impossible, then all of them are impossible.

 

[icerat]

That sequence is just as possible as any other unique sequence of 100 coin flips.

which would be 1:2^100, but only before you start tossing the coins, and it's the same for *any* sequence of a 100 coin flips. And clearly it's not impossible to get a sequence of 100 coin flips.

where people get confused is not understanding that the probability of throwing a head, after having thrown 99 heads in a row, is 1:2

 

[Alan]

It's also entirely possible to bowl a 300 game, though very much less likely than any other bowling score.

Are you suggesting that this is also the case with coins?

You would be wrong. Each flip is 50/50. Each flip has an equal chance of being anything. This is not the case in ten pin bowling.

 

[AdMan]

It's also entirely possible to bowl a 300 game, though very much less likely than any other bowling score.

Yes, bowling a 300 game is much less likely than any other score, because it's not random. It involves skill.

With tossing a coin 100 times, getting 100 heads in a row is no less likely than getting another similarly unique sequence, even if it looks like a random mix of heads and tails.

 

[Myriad]

Here's another way to clarify the issue:

As explained already, any specific sequence of 100 flips is equally possible, so of course they're all possible.

What is not possible is to predict the sequence that will occur, with any practically significant degree of confidence.

So, it's absolutely not possible to confidently predict HHHHHHHHHHHH.... Nor, of course, HTTTHTTHHHTH... or any other specific sequence.

But the only absolute impossibility is of confidently predicting. That impossibility does not rub off onto the possibility of any specific sequence occurring. And thus, it does not preclude the possibility of the sequence you predict occurring; or in other words, the possibility of correctly predicting. Those are merely astronomically unlikely.

Respectfully,
Myriad

 

[TubbaBlubba]

It's discussion like these that sometimes make me question the sanity of some of the most brilliant posters here.

 

[Alan]

What is your position?

 

[Piggy]

Thanks, Alan, for starting the thread.

It's nearly midnight here, way past my bedtime, and I have appointments in the big city tomorrow, so I have to do a drive-by (sorry, Sol) but I now have this thread on my subscription list.

First I'll say that I have no objections to the stats regarding the range of possible outcomes of a system which produces all mathematically possible configurations of T/H for a given series length.

As far as the gambler's fallacy, just to give a kind of preamble to my way of thinking, let's rephrase it this way....

Suppose you're betting on a run of 100,000 fair coin tosses that has already occurred. At random, the result of 50,000 of those tosses are revealed.

As it turns out, 75% of the revealed results are heads, and only 25% are tails.

What odds are you willing to take that the remaining 50,000 will be evenly split among heads and tails? Would you accept even money on a bet that 25,000 (+/- 2,500) of the unrevealed flips ended up heads? Or would you be inclined to bet that the majority of the unrevealed flips were tails?

In other words, would you bet that the one-off event (the revealing of a random selection of results) was the abberation, or would you bet that the extended series of events resulted in a significant abberation?

 

[nimzov]

Given that you have, for example, 10 coin tosses, and the results are either H or T, the sequence HHHHHHHHHH is no more unlikely than any other, such as HTHHTHHTTH.

In theory. But in practice the first sequence would indicate a higher probability of an unfair coin than the second sequence.

 

[Piggy]

In theory. But in practice the first sequence would indicate a higher probability of an unfair coin than the second sequence.

I'll elaborate more later, but my position is that the statement you're responding to is inherently flawed.

The real question we need to be dealing with is this: Given a supposedly fair toss, is a completely smooth results-space more or less likely to indicate cheating than is a non-smooth results-space (regardless of the exact configuration of the non-smooth space)?

By analogy, let us consider the task of determining whether we are in the ocean or a swimming pool by means of examining the behavior of the water within a nearby radius.

If we detect low turbulence and no tidal effect, then we are either in a swimming pool, or in an unusually stable sector of ocean.

Suppose that the circumstances do not change over the course of 2 hours.

Assuming that we know we are somewhere in the real world, do we conclude that we might be in the middle of an unprecedented anomaly, or that we're in a swimming pool?

If we listen to the logic of the post you're responding to, we simply count the "extremly unlikely" low-turbulence/no-tide scenario as one of a large number of equally unlikely possible configurations.

But if we use our faculties of reason, we will realize that there are a large number of configurations that conform to "we are in the ocean" and a much smaller number of configurations that conform to "we're in a pool".

So this business of claiming that a run of 100 heads is equally un/likely as any particular scenario of mixed tails and heads is a red herring.

The important feature to recognize is that a vast number of mixed heads/tails configurations will be typical of the results-space of a long series of fair coin tosses, while a long series of only heads or tails is typical only of a rigged system.

The appeal to the supposed equal likelihood of any one particular configuration is a misapplication of statistics to the actual situation on the ground.

 

[sol invictus]

As far as the gambler's fallacy, just to give a kind of preamble to my way of thinking, let's rephrase it this way....

Suppose you're betting on a run of 100,000 fair coin tosses that has already occurred. At random, the result of 50,000 of those tosses are revealed.

As it turns out, 75% of the revealed results are heads, and only 25% are tails.

What odds are you willing to take that the remaining 50,000 will be evenly split among heads and tails? Would you accept even money on a bet that 25,000 (+/- 2,500) of the unrevealed flips ended up heads? Or would you be inclined to bet that the majority of the unrevealed flips were tails?

That question is impossible to answer (at least for me) until you tell us more precisely what the circumstances are. Specifically, exactly how do we know that the coin is "fair"?

 

[Vorpal]

In other words, would you bet that the one-off event (the revealing of a random selection of results) was the abberation, or would you bet that the extended series of events resulted in a significant abberation?

It's not a particularly relevant question, because as in the other thread, multiple people have tried to make you see the explicit distinction between
(1) whether for a fair coin, all sequences are equiprobable, and that a particular high-run sequence is just as likely as any other particular sequence when you're predicting before the fact, and
(2) whether in observing a coin that gives a very high run of heads, it is reasonable to conclude that the coin is not fair (which is actually simpler than the situation you've just posited).

You seem to be rhetorically arguing that the answer to (2) is a resounding 'yes'. I don't think anyone has said otherwise. Moreover, that answer is perfectly consistent with answering 'yes' to (1).

For a fair coin, no high-run sequence is any way more special than any other sequence. What can be considered special is the event of getting some (nonspecified) high-run sequence, i.e., the set of high-run sequences is much less numerous than the set of sequences without high runs.

You clearly recognize this:

Ah, but that's an improper comparison. ... The proper comparison would be TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT..... versus everything else, not versus a single alternative chosen at random.

And then Sol followed up in #123:

There are two questions that need to be separated here.

Q1) Is a sequence of 100 heads on a fair coin possible?
A1) Yes. The claim that it isn't can be immediately disposed of by noting that said sequence is no more unlikely than any other, and some sequence must always result.

Q2) Is a sequence of 100 heads on a supposedly fair coin more likely to be the result of a truly random fair sequence or of cheating?
A2) Cheating, almost certainly.

No one disputes your claim about concluding unfairness, at least in that particular case. The dispute is entirely based on you not admitting that (1) is a completely correct statement in its own right that's also consistent with (2).

 

[Little 10 Toes]

The problem with your ocean vs pool analogy is that I could be in a lake. Is that a pool or the ocean. I could be standing in a puddle and fit every other criteria. The other problem is that you're trying to use an analogy when one isn't really needed.

If I flip a fair coin, and the first two results are HH, what likelihood is the third flip to be H? It's the same chance that it will come up to be T.