Oh, what the hell, i'll give it a shot too:

@Synapse: you've gotten many different versions of the correct answer, but you're still not quite there. So here are a few thoughts:

1) This is NOT physics ... it is just math. To bring the physics of how a dice is rolled is to miss the mathematical point. If you agree that the chance of getting heads is 1/2 and that the chance of getting a 6 on a die is 1/6 then good, we're already ignoring the physics involved. It's just math! don't complicate it unecessarily.

2) There is no difference between a die and a coin. There is only a difference between the number 2 and the number 6. consider a die a 6-sided coin,or a coin a 2-sided die. It doesn't matter. as long as you agree with the statements in point 1)

3) There is absolutely no difference between flipping 2 coins at once, and flipping one coin twice. Or rolling 1000 dice once vs rolling one die 1000 times. They're independednt events in both cases.

4) This has absolutely NOTHING to do with "continuous" probabilty (probability distribution?). The reason people start with coins and dice when they're learning probability is because it keeps it simple. Dont go claiming it's continuous probabilty when you can't get the simple stuff down.

Now lets do a thought experiment. If you say that the odds of getting 2 heads when you flip 2 coins (or the same coin twice - the math can't tell the difference and either can the coins) is 50/50, then you must also agree that flipping a coin 10 times, the odds are still 50/50. If not, then where does it start NOT being 50/50? 3? 4? I'll assume that you agreed that it is 50/50 to get all heads when throwing 10 coins up into the air. That means that the possibilty of all heads is just as likely as all other possibilities combined. And surely the math can't tell the difference between Heads and tails, so the same can be said for the case of getting all tails - 50/50. Just as likely as all other possibilities combined. But this implies that you have 50% chance of getting 10H and 50% change of getting 10T - that's 100% so there can't be any other possible outcomes! Therefore you must either conclude, either:

A) you can never get a mixture of heads and tails when throwing 10 coins into the air, or
B) you're not doing the math correctly

Probability is nothing more than favorable outcomes divided by total possible outcomes. That's it. From there it's just a matter of finding the right math to solve for those 2 quantities. But with coins and dice, you can just write out all possibilities and circle the favorable ones. if you do this with coins, because the numbers are smaller, you'll see very quickly the number of times you throw it you have double the possibilities.

One Coin - two possibilities:
H
T

Two coins - 4 possibilies. Each of the possibilities from above are still there so the second can be H or T with each of the above possibilities:
HH
HT
TH
TT

3 coins - 8 possibilities for the same reason above
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT

If you can agree that this reasoning is correct (it's now just boiled down to reasoning, there is no math OR physics involved anymore), then you can see that there is only one way each for all heads or all tails to occur. So the probabilty is going to 1/number of possible outcomes. Now we need math again - but it's simple math.

for 1 coin, there were 2^1 possibile outcomes so Prob = 1/2
for 2 coins there were 2^2 possible outcomes so Prob = 1/4
for 3 coins there were 2^3 possible outcomes so Prob = 1/8
for 10 coins there are 2^10 possible outcomes so Prob = 1/1024

Do you see that yet? Do this with dice and everything, the reasoning, the math, etc. is exactly the same. The number of possibilities per roll is the only thing that changes. It changes from 2 to 6, that's all. If you had a 4 sided dice or a 20 sided dice it still the same.