I alternatively toss a fair coin and a fair die until I, either toss a head or throw a ‘2’. 1f I toss the coin first, the probability that I throw a ‘2’ before I toss a head, is

H : tossing a head, P(H $=\frac{1}{2}$event of tossing a 2 with die, P(A) = $=\frac{1}{6}$
E : tossing a 2 before tossing a head
$\begin{array}{l}\mathrm{P}\left(\mathrm{E}\right)=\mathrm{P}(\stackrel{ }{\overline{\mathrm{H}}}\cap \mathrm{A}\text{ or }\{(\stackrel{ }{\overline{\mathrm{H}}}\cap \stackrel{ }{\overline{\mathrm{A}}})\text{ and( }\overline{\text{H}}\cap \mathrm{A}\left\}\text{ or.... }\right)\\ =\left(\frac{1}{2}\cdot \frac{1}{6}\right)+\left(\frac{1}{2}\cdot \frac{5}{6}\right)\cdot \left(\frac{1}{2}\cdot \frac{1}{6}\right)+\dots ..=\frac{1}{12}+\frac{5}{12}\cdot \frac{1}{12}+\dots \mathrm{\infty }\\ \mathrm{P}\left(\mathrm{E}\right)=\frac{\frac{1}{12}}{1-\frac{5}{12}}=\frac{1}{7}\end{array}$