A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads?

Question

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?

Answer 1

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.

Answer 2

Hint:

1. Fair coin $\implies$ Probability of tails occurring more $=$ probability of heads occurring more $= p$, say.

2. Probability of exactly equal number of heads and tails $=1-2p$. Can you find this one?