TITLE: A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads?
QUESTION [6 upvotes]: I'm studying for a probability exam and came across this question. I watched the video solution to it but I don't really understand it. I was hoping someone could explain this problem to me. Are there different ways to go about this?

REPLY [37 votes]: Hint:
The probability that an equal number of tails and heads appear is $\large{{2k \choose k} \frac{1}{2^{2k}}}$
The two remaining outcomes (that there are more heads than tails or more tails than heads) are equally likely.

REPLY [13 votes]: Hint:

Fair coin $\implies$ Probability of tails occurring more $=$ probability of heads occurring more $= p$, say.
Probability of exactly equal number of heads and tails $=1-2p$.  Can you find this one?