1. @Dragan - the gambler's fallacy argument is a great way to understand the issue, one I hadn't heard of. I had been thinking about how the problem here is an inaccurate cognitive heuristic wherein probability is almost anthropomorphised as a sort of mechanism or divine hand that 'corrects' improbable runs of events (rather than any statistical paradox or problem). The gambler's fallacy is a good explanation of this.

2. I came here believing i would find some answers to help me in an upcoming exam, i didnt think i would actually get interested in the topic!
thanks alot very interesting question and answer!

3. These are good answers to my question. But... I am still not satisfied. The Gamblers fallacy is simply a mathematical rehashing of the previous answers. The flip is always 50/50. I know that. But the fallacy still seems like a cop out of the real issue.

You know things even out in the long run because they always have. That is what we observe, which lends to the idea that it is what we expect to happen. Flipping 10 heads in real life will most likely be followed by a tails. The chances before the 1st flip of 11 heads are slim, so even if you are present during the 11th flip, you can assume you are also present back at the 1st flip expecting that tails on the 11th. This itch has not really been scratched yet.

Thanks for all who have the endurance to keep this discussion moving.

4. Originally Posted by djarvis

But...I am still not satisfied.
And, no matter what anyone says...you will never be satisfied. (Right?)

5. Originally Posted by djarvis
Flipping 10 heads in real life will most likely be followed by a tails.
No, it won't. Your assumption that this is the case is what you need to reconsider. There really isn't a paradox here.

6. yes, Dragan, I certainly can, but I figure this is a great place for drawing out such good discussion. Maybe through this discourse I can solidify this cloudy idea and in the future it won't be an issue.

CowboyBear, You are right, my assumption that the likelihood changes is wrong, but something doesn't feel right. I guess that is why the gamblers fallacy exists (thanks Dragan).

This is the most difficult part for me: The probability of flipping 2 consecutive heads is .25. That means after the first flip, if it was heads, the probability of another heads is .25. But it isn't really .25 because the first flip has already occurred. So this seems to mean that we can never predict the next event, but we can always predict the event after that, given potential outcomes of the next event? This means that the event after the next event is related to the next event, until we get to it, and then it becomes independent...

7. I understand your question, djarvis. It makes me think of the movie Beautiful Mind where Russell Crowe says to his friends at a bar before approaching a girl: "My odds of success dramatically improve with each attempt". Where I take this to mean that he knows he may get shot down, but getting shot down will inevitably improve his chances - in the long run.

8. Having said that - if there were lets say 1-million people each of whom flipped 10 heads in a row, each of whom are to flip an 11th coin - what kind of distribution would you expect to see from those million 11th flips? The fact is, they would probably be pretty close to 50/50.

9. Let's say we've tossed a coin and get three heads in a row...

HHH...

What is the probability that the tenth toss is also heads? The answer is 1/(2^4) = 1/16:
HHHH

What is the probability that the tenth toss is tails? The answer is the same - 1/16:
HHHT

The probability of getting either one of these sequences is identical. In fact, the probability of getting any arrangement of heads and tails for four throws is as above:

HHHH
HHHT
HHTH
HTHH
THHH
HHTT
HTHT
THHT
HTTH
THTH
TTHH
HTTT
THTT
TTHT
TTTH
TTTT

If we look at all possible permutations, the probability of getting at least one T is 15/16, however. These are two very different things: P(T) on the fourth toss vs. P(T) on any toss.

Also...

"Flipping 10 heads in real life will most likely be followed by a tails..."

...is absolutely wrong. I think this is the heart of your problem. If you were to flip a fair coin and keep track of the results for every 11-toss run in which the first 10 tosses were heads, I suspect you would find that P(H) and P(T) are both about 1/2.

We can test this via simulation, but it might be easier just to get a coin and try this for 3 tosses. If your supposition holds, then every time you toss a coin twice and get HH, the chances of the third toss being T should be higher than for H. So go ahead - toss a coin a bunch of times, and keep track of how many times you get HHH vs. HHT.

10. May I ask you some questions?
Do you think that each flip of coin will affect the probability of getting a head
or a tail in the next flip (e.g. physically)?
If your answer is no, then we may assume it is an independent sequence.
Proceed to the following question.

Do you think that I can change the odd of the fair coin verbrally? (kidding)
I think no human being can do that with a fair coin.
Then we may think of the following four situations:
Suppose you are ready to flip a coin which you believe is fair.
Before you filp the coin, I tell you that I have flip the coin 10 times before,
a) and I got 10 heads
b) and I got 10 tails
c) and I got 5 heads and 5 tails
d) and I do not want to tell you what I got in those 10 flips

Do these information matters you?