Why or Why not? Statistically, one would think you only have two choices.

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This follows a binomial distribution. The probability of getting exactly 50 heads from 100 tosses is (100C50) (0.5)^50 (0.5)^50 = 0.0795892.

What this means is that if you repeated this trial 100 times you'd expect to get exactly 50 heads in 8 of those trials. Although this seems small, it is the most probable individual outcome.

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Edit: Hmm, I see some nice pages on the internet that discuss this in more detail with physics and the realities of actually flipping a coin (rather than an arbitrary 50 / 50 split generator).

@Klena: Surely you mean the other way around? When you flip a coin only 100 times, it is far more likely that you'll end up with 51 - 49. But when you flip the coin a large number of times, you'll approach and go to 50 - 50. The smaller the number of flips, the less likely it will be exactly 50 - 50. If I flip only 4 times, it may very well be 3 - 1 or 4 - 0 but when I flip a million times, it will be a lot closer or exactly a 50% - 50% split of heads and tails.

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Assuming a fair coin, exactly 50 heads and 50 tails is more probable than any other possibility, but it is lower than the sum of the other possibilities.

That is to say, the odds of 50h/50t are greater than the odds of 51h/49t and the odds of 50h/50t are greater than the odds 49h/51t, but the odds of 50h/50t are lower than the odds of 51h/49t OR 49h/51t.

In fact, the odds of exactly 50/50 in 100 flips are fairly low, even though it's the most likely single outcome.

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Just to correct you: the odds of getting 50h/50t is exactly the same as 51h/49t. There are 101 outcomes (0h/100t, 1h/99t, 2h/98t etc.) and they're all equally likely of happening; about 0.0099%.

This will depend on how you flip the coin and the coin itself. Certain part of the coin may be heavier than the rest which could affect the outcome. The way in which the coin was flipped could also be used to manipulated the result. Take the height of the coin flipped for example, changing the height changes the number of times the coins flip-over before it reaches the floor or your hand. If you have ever pushed your buttered toast of the table, you will notice that it usually falls buttered-side down. One of the reasons for this phenomena is because of the height of the table which is too low for the bread to make a complete turn before reaching the ground.

If you are skillful, you could get use to the coin you are flipping and thus be able to control the outcome most of the time.

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rabid puma has got this correct.

intuitively, the reason 50/50 is most likely is because there are more ways to get 50 heads and 50 tails.

for example, there is only one way to get 100 heads. to get heads every time. However, there are 100 ways to get 99 heads and 1 tails (THHHHHH... ; HTHHHHHH... ; HHTHHHHH...; etc)

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Q.I actually answered this question dring a show. If you flip a coin 100 times, it will be equally 50-50.

However, if you flip the coin a larger number of times for a better data spread, it does actually turn out that it will be 51-49 on one answer.

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Theoretically it has to be 50-50 (I bet you've heard: "probability has no memory" before), the 51-49 that Klena mentioned are due to the slightly unbalanced weight of many coins where actually a side tends to be heavier (if you have e.g. George Washington's face in the middle of one side of the coin while the statue of liberty is on the right side of the coin there's not the exact same amount of gravitational force and of course friction acting on the coinage metals involved - they too aren't distributed evenly if you observe it from a molecular point of view) but nevertheless it turns out that most coins favor one side a tiny bit more thus when throwing it only 100 times you won't notice this small "misbehavior" but when thrown e.g. a billion times, suddenly the effect would become slightly visible).

But all this is just based on calculations and brain-teasers of some bored physicists and actually no one has tested it so far but chances (oh here good old probability strikes again) are that at least a 51-49 pattern would emerge i.e. be observed ;)

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