Probability - How does it work?!

^yes, each specific sequence is equally likely but it depends on whether the order has to be considered.

The problem (as usually stated) is what are the chances of flipping a head 100 times in a row. Rather than in an arrangement of 100 coins what percentage of possible arrangements feature XXXXX (XXXX being whatever you are measuring).

As useless was pointing out, many individual sequences give the same number of heads and tails if that is all you are interested in.

So with 4 coin tosses the possible outcomes are:

HH --- 1/4
HT --- 1/4
TH --- 1/4
TT --- 1/4
Therefore a head and a tail (order not important) = 1/4 + 1/4 = 1/2

They are all equally likely but if the order is not important it is 2 times as likely to get a head and a tail.

Well, strictly speaking you are not twice as likely to get a head + tail (on a two coin toss) when accepting combinations in place of sequences, the results aren't influenced by the rules you are using, only your interpretation of the results, but I understand the point you are making.

If a run of 'x' coin tosses was repeated infinitely many times each possible sequence would occur

Probably, but not necessarily.

I once had a long ongoing argument with a whole studio (around 20 people) where I worked (this was when the lottery first started so probably around 1995) - based loosely around this problem, basically a guy on the radio had said, during a piece on the lottery, that it was a mistake to choose all even (or all odd numbers) as those combinations are much rarer than mixed even and odd sequences.

It literally took me a month to convince people that the idea was nonsense, so strong was people's instinct that certain numeric sequences are significant in a system like this, it actually got quite funny at times, hilariously so, the whole thing ended in me actually making 49 little balls with 'Carry On' stars replacing the numbers to try and break the connection (and significance on sequence) people can erroneously place on numbers.

I got them to argue the case that:

Hattie Jacques, Sid James, Jim Dale, Terry Scott, Windsor Davies . . . (etc etc)

. . . was more (or less) likely to come up than . . .

Kenneth Williams, Charles Hawtrey, Joan Sims, Barbara Windsor, Leslie Phillips . . . (etc etc)

. . . and to explain the system by where one Carry On film star was more likely to be chosen than another, did the system need to know who was on the ball for instance, if we pained over all the 'Carry On' stars faces with green paint, obscuring them, would Hattie Jacques be still more (or less) likely to be chosen than Kenneth Williams.

I eventually became victorious, proud in having brow beaten 20 heathens into logical submission, but all the time I was slightly saddened knowing that while I had spent the best part of a month playing with my balls the guy on the radio was probably going out, getting pissed, getting laid and enjoying himself.

[/weeps]