JEE Advanced 2023 · Paper 2, Mathematics question 2

Let $p=\frac13$ (head) and $q=\frac23$ (tail).

Shape of a run ending in heads. The experiment stops with head exactly when the tosses alternate and then end in HH. Before the final H there is an alternating string ending in H:
$$H,\ TH,\ HTH,\ THTH,\ \dots$$
Their probabilities are $p,\ qp,\ p^2q,\ p^2q^2,\dots$

Sum.
$$\sum=p(1+q)\left(1+pq+(pq)^2+\cdots\right)=\frac{p(1+q)}{1-pq}.$$

Answer. Multiply by the final $p$:
$$P=\frac{p^2(1+q)}{1-pq}=\frac{\frac19\cdot\frac53}{\frac79}=\frac{5}{21}.$$