Problem
A fair coin ($P(H) = P(T) = 0.5$) is independently tossed $400$ times.
Is it correct to say that the probability of getting from $0$ to $160$ heads is the same as the probability of getting from $240$ to $400$ heads?
On the one hand, the answer seems 'no' because the number of heads are different.
On the other hand, if we take the first numbers of heads, from $0$ to $160$, it means that at the same time I want from $240$ to $400$ tails and because the coin is fair, it is the same probability as in the second case with $240$ to $400$ heads.
So is my initial statement correct?
I appreciate any help.