A question in probability by a non-mathematician:
A fair coin is tossed $2N$ times. Is it unlikely that we get exactly $N$ heads and $N$ tails?
From one side, this must be the most likely result! But intuitively, if someone reports to me that they threw 2,000,000 coins and got exactly 1,000,000 heads and 1,000,000 tails I would be a little suspicious.
If this really is suspicious, how does it compare to biased results? Obviously, 2,000,000 heads is a much more suspicious result. But, for what value of $K$ is $N+K$ heads and $N-K$ tails as suspicious as $N$ heads and $N$ tails?
Equations are very welcome, but please try to provide some aid in understanding them.
ADDED: Henning Makholm's answer helped me understand better what I want to ask: I always assume a fair coin and would like to compare the probability of the split N/N with the probability of getting either more than more than N+K or less than N-K heads. By ncmathsadist's answer , the probability for the N/N split goes like 1 over the square root of N. What about the probability to get more than either more than N+K or less than N-K heads? For what value of K (approximately) does it equal to 1/sqrt(n pi)?
CLARIFICATION: If f(N) is the probability to get an N/N split in 2N tosses of a fair coin, and g(N,K) is the probability to get more that N+K heads or less than N-K heads in 2N tosses of a fair coin, then I am asking for a formula which, given N, approximates the values of K for each f(N) is closest to g(N,K). This is a formula that takes N and returns K so we can call it h(N)=K. Which kind of function is h? My guess from a little experementing with the online software in Brian M. Scott's answer is that the function h is apporximately the SQRT function, multiplied by some constant, or something like that.