In a hockey match, both teams $A$ and $B$ scored the same number of goals up to the end of this game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team $A$ is asked to start, then probability of $B$ winning the match is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\frac{5}{11}$ ##
$B$ will win in second attempt or fourth attempt or sixth attempt or so on
$∴P(B \text{ winning}) = P(\overline{A}B) + P(\overline{A} \ \overline{A} \ B) + P(\overline{A} \ \overline{A} \ \overline{A} \ \overline{A} \ B) + \cdots$
$= \frac{5}{6} \cdot \frac{1}{6} + \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6} + \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6} + \cdots$
$=\frac{\frac{5}{36}}{1 - \frac{25}{36}} = \frac{5}{11}$