Probability of first player winning

Question

Two players take turns to toss a coin; the winner is the first to toss a head. What is the probability that the first player to toss the coin wins?

Answer 1

The probability that the first time a head is flipped is on the $i$th turn is $(1/2)^i$. Thus, the probability that the first person wins is

\begin{align} (1/2) + (1/2)^3 + (1/2)^5 + \dots \end{align}

Can you recognize this as a geometric series?

Answer 2

Let's call $p$ the probability the first player wins

Assume first player tosses heads the first turn then he wins. If that is not the case then he tosses tails, hence the probability he wins is if the second loses which is equal to $1-p$ (the roles have been reversed)

Put it all in an equation $p= \frac{1}{2} \cdot 1 +\frac{1}{2} \cdot (1-p) \Rightarrow p=\frac{2}{3}$