FleaCircus

My question is a little involved, but here goes:

Let's say you're flipping a coin--you have two possible outcomes, heads and tails. The likelihood of heads coming up once (in one toss) is 1/2, twice (in two tosses) is 1/4, three times (in three tosses) is 1/8.

So we can generalize that the chance of heads coming up n times in n tosses is 1/2^n.

But how can we generalize the chance of heads coming up any x times in n tosses, where x<n? For example, what's the probability of any 7 of 10 tosses coming up heads?

Next, let's say that I have a friend who claims he's psychic, and can predict the outcome of a coin toss. If he gets 7 out of 10 tosses right, that's pretty good, but the generalization above gives us the probability of such an outcome happenning by random chance alone. At what point does this potential for an outcome by chance alone become statistically negligible?

I know that, even with a million tosses, there's still an outside chance that he could get them all correct by randomly guessing. But it's so slim a chance that, if he could get that many right (and especially if he could repeat it), then I'd presume that chance alone did not dictate the outcome.

And then I'd wear a tinfoil hat so that he couldn't read my mind.

Edit: I'm not trying to prove/disprove psychic abilities, I'm just using it as an example.

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