Question

Sol. [Key Concept: Geometric series for whoever first throws the target sum on the dice.]

$P(A) = 6$ or $\overline{6}\ \overline{7}6$ or $6\overline{7}\overline{6}\overline{7}6$ or $\ldots\ldots..\infty$

$$\begin{matrix} & = \frac{5}{36} + \frac{31}{36} \times \frac{30}{36} \times \frac{5}{36} + \left( \frac{31}{36} \times \frac{30}{36} \right)^{2} \times \frac{5}{36} + \ldots.\infty \\ & \ = \frac{5/36}{1 - \frac{30 \times 31}{(36)^{2}}} = \frac{5 \times 6}{216 - 155} = \frac{30}{61} \\ P(\text{ }B) = & \frac{31}{36} \times \frac{6}{36} + \frac{31}{36} \times \left( \frac{30}{36} \times \frac{31}{36} \right) \times \frac{6}{36} + \frac{31}{36} \times \left( \frac{30}{36} \times \frac{31}{36} \right)^{2} \times \frac{6}{36} + \ldots\infty \\ & \ = \frac{\frac{31}{36} \times \frac{6}{36}}{1 - \frac{30}{36} \times \frac{31}{36}} \\ & \ = \frac{31}{61} \end{matrix}$$