UNSW MathSoc Championship Q10

Let $p=\mathbb{P}(TT|H)$ denote the probability that the game ends with two successive tails given the first toss results in a Head, Similarly define other events. Define $q=\mathbb{P}(HH|H)$ . Then, due to symmetry, we have $p=\mathbb{P}(HH|T), \hspace{10pt} q=\mathbb{P}(TT|T), \hspace{10pt} p+q=1.$ Conditioning on the first outcome, we have $p = \frac{1}{2}\times 0 + \frac{1}{2}q$ . This gives $p=\frac{1}{3}, q=\frac{2}{3}.$