TOSS Even

After the coin is thrown an odd number of times alternating between heads and tails, the probability of the last throw being either heads or tails is even. Then the following could occur (first coin being at the end of an odd number of throws)

$a: {H,T,T}$
$b: {H,H}$
$c: {T,H,H}$
$d: {T,T}$

with events $b, d$ being twice as likely as events $a, b$ , so the probability of an even number of throws is $\dfrac{2}{3}$

The Sample Space is $\Omega=\big\{\text{HH,HTHH,HTHTHH,}\ldots\big\}$ or $\Omega=\big\{\text{TT,THTT,THTHTT,}\ldots\big\}$ .

So, the required probability is $2\left[\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^4+\left(\dfrac{1}{2}\right)^6+ \ldots\right]$ $=2 \cdot \dfrac{1}{4} \cdot \dfrac{1}{1-\frac{1}{4}}$ $=\dfrac{1}{2} \cdot \dfrac{4}{3}$ $=\boxed{\dfrac{2}{3}}$