Out of Pod Experience

He's talking about the concept of a perfectly fair coin being tossed by an unbiased thrower, almost exactly equally likely to end up heads or tails, and having all results of the type "landed on the side" or "lost somewhere" cropped out.
If you prefer, imagine instead a hypothetical perfectly random bit generator (equally likely to generate a 1 or a 0 on any interrogation), and call every 1 "heads" and every 0 "tails".
Because in the end, that's what we've been doing anyway.

That's not the problem in his logic anyway (at least not as long as he doesn't mention how actual randomness can't exist or anything like that).
It's the other conclusions he occasionally proclaims that are actually problematic.

All that is irrelevant.
The only relevant thing is that you CAN get ANY finite sized streak of heads or tails as long as you just keep on making throws as long as you like, even until after the universe would have normally ended if you pick a ridiculously high streak length.
:P

30 heads ? Sure, can be done by an actual person in a lifetime.
40 heads ? Can be done by a planetary population in a lifetime.
50 heads ? Can be done by a supercomputer simulating throws for a few years.
60 heads ? Can be done by all computers in the world simulating throws for a few years.
70 heads ? Can be done by all computers in the world in a few generations.
100 heads ? Can be done probably before humanity ends if we manage to get nifty enough computers.
1000 heads ? You might manage to do it before the universe ends if whatever exists wants to keep on trying.
100000 heads ? You could do that if you could have access to a few universes that do nothing else but simulate it.
1 billion billion billion heads ? MORE UNIVERSES PLEASE and it's done.
and so on and so forth

As long as it's NOT IMPOSSIBLE to throw heads in any given throw for some reason, there's always a chance to throw it, so you can always make a streak that's 1 longer, it just takes longer and longer until you actually get it for each additional 1 to the length.
Unless you claim that after a certain heads streak length the coin all of a sudden has to flip tails with no chance to flip heads, that's the inescapable conclusion.
And you already admitted that you don't deny any individual flip remains at 50:50 chance.
So there you go. End of story.

A Question for Akita the math wiz. I am trying to figure out how you calculated the part where you said there is a 3:256 chance of flipping 7 heads in a row in 8 flips.



If:
The odds of flipping 7 heads in a row is 2^7
and the odds of flipping 20 heads in a row is 2^20


Then:

(2^7) / (2^20) = (1:128) / (1:1,048,576) ?

Thus the odds of flipping a 7 heads streak in 20 flips is 8,192 / 1,048,576 ??

Thus the answer ends up being .00781?


Is that even close to right?

From your OP:



long streaks RESULT in a diminished probability of continuation. That means that the longer your streak, the lower the probability of it continuing. The ONLY way that can occurs is by modifying the odds of the individual coin toss. Thus, every coin toss would be influenced by previous coin tosses.

As you completely ignored the examples I gave around this, I'm going to assume you had nothing to refute my points.

You want to know something that throws all the statistics we've bantered about right out the window and makes these streaks far more likely than you think? We've been working on the basis of sets. For example, suppose we're trying to find a string of four 1s. The math suggests that we have 2*2*2*2 possible combinations, and so our odds of finding four consecutive 1s would be 1:16. That is obvious when you type out all the possible combinations:

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

However, when you remove the breaks and treat it as a single large set, you get:

0000000100100011010001010110011110001001101010111100110111101111

You find four instances of "1111" in the sequence. Yes that is a non-random sequence, so here's a random one I just generated:

0001000111011110000101110001111001011100110011011000101001001010

There are two sets of "1111" in that. However if you group it into sets of 4, the sequence never occurs. "0001" occurs four times, though. The point is, even the statistical predictions we have relied on in these threads are very conservative.

You talk about our brains perceiving things which statistics and mathematics might not. Our brains also see Jesus on a piece of toast. We're evolved to look for patterns to the point that we see them even where they don't exist. We assign significance to certain things because they *appear* orderly, even when they don't. 01010101 is not a significant sequence of coin flips, it's just one of 256 possible outcomes. It's your own mind that tricks you into thinking it's significant because it stands out above 011000111 and 10010011.

When you reframe the argument to a single person's lifetime, of course the odds are so strongly against 1,000 consecutive heads. However, if you've just watched 999 heads flipped, the odds of the next toss making 1,000 are *still* 50:50. Nothing is going to change the odds of that individual coin toss.

This whole thing started because you seemed to be asserting that the odds of a single coin toss changed based on previous coin tosses, making it increasingly unlike that a streak would continue. That's simply not the case. The problem is that the longer your streak the more rare it becomes, and only one out of every two will get longer. So if you ever do get 100 heads in a row and then get tails, you're going to have to basically start all over trying to get that 100 again and see if you flip heads at 101. However long it takes to get a particular streak, double that number for the next highest. That's what we've been trying to convey to you from the start of this whole mess.

It seems you're backing off that stance now and framing the argument in such a way as to state "well, we'll never see it." Which is of course almost certainly true, as the odds of any given combination of such length are so tiny that it doesn't matter what sequence of 1,000 random bits you choose, you've probably never seen it.

8 flips means 2^8 = 256 total possible combinations
There are exactly three possible combinations of 8 flips which contain at least 7 heads in a row : 7h+t, t+7h, 8h.
So, odds of 7 heads or more in 8 flips is 3:256 = 1.171875%


Nope, not even close to right.

You'd need to count ALL possible combinations of length 20 which contain at least 7 heads and divide by 2^20 to get the exact answer.
This is quite painful to calculate with exact precision.

For instance, there's:

* 2^13 combinations that start with 7 heads, (8192, those you did take into account)

* 2^12 combinations that start with t+7h, (4096 more)

* 2^11 combinations that start with h+t+7h (2048 more,
* 2^11 combinations that start with t+t+7h (2048 more)
= 4096 more in this group

* 2^10 combinations that start with t+t+t+7h (1024 more)
* 2^10 combinations that start with t+h+t+7h (1024 more)
* 2^10 combinations that start with h+t+t+7h (1024 more)
* 2^10 combinations that start with h+h+t+7h (1024 more)
= 4096 more in this group

* 8 sets of 2^9 combinations that start with 3anything+t+7h (8*512=4096 more in this group)

* 16 sets of 2^8 combinations that start with 4anything+t+7h (16*256=4096 more in this group)

* 32 sets of 2^7 combinations that start with 5anything+t+7h (32*128=4096 more in this group)

* 64 sets of 2^6 combinations that start with 6anything+t+7h (64*64=4096 more in this group)

* 128-1=127 sets of 2^5 combinations that start with 7anything+t+7h ; 128-1 because we needed to exclude 7anything=7h because it would be a duplicate of one of the combinations listed first, but all the rest are ok (127*32=4064 more in this group)

...and this is where it starts to get complicated because more and more possible sets would be duplicates of previous sets, so let's ignore everything from that point onwards for now and say everything might be a duplicate.

That would be so far at least 8192 + 4096 * 7 + 4064 = 40928 sets out of 2^20 that have at least 7 heads in a row.
40928/1048576 = 0.039031982421875, so there's at least a 3.9% chance to roll 7 heads in a row out of 20 flips.

The total sum of throws will balance out over long periods of time, but in no such order as to benefit the gambler. Now imagine if you rolled five 1’s in a row. Now estimate how long it will take until you stop throwing 1’s within reason during the time allowed.

Relativity can be applied to anything involving an observer by the way.

The odds are in favor of my very next roll breaking the streak. However, it's entirely possible (and I've personally seen it happen) for that streak of 5 to turn into 10 before it breaks. That's the thing...with a fair toss, the outcome of an individual selection can't be predicted. In this instance it's "reasonable" to expect that streak to break every time you roll, but the odds of rolling 111111 are exactly the same as rolling 123456 or any other combination. I simply happened on that one that stands out the most.



Actually, relativity is quite specific in scope. It demonstrates that time and space are perceived differently by observers travelling at different velocities. It has nothing to do with probability.

To emphasize Floppie's point, "streaks" do NOT actually exist. A streak is just a sequence that looks special to the observer (i.e. all heads, all tails, alternating heads/tails, etc..) Mathematically, every sequence of numbers is a streak. If everything is a streak, then nothing is a streak.

analogy: By implying that a streak must be broken, you're guilty of anthropomorphizing a sequence of numbers.