Question

Ram throws a fair six - sided die untill six appears for the first time. Independently Shyam rolls the die until six appears for the first time. Now, which of the following options holds good?

• A. Probability that Ram gets six as result on third throw is $\frac{25}{216}$
• B. Probability that Shyam gets six as result on or after third throw is $\frac{25}{36}$
• C. Probability that Shyam gets six as a result on or after third throw is $\frac{25}{211}$
• D. Let $\alpha$ and $\beta$ be the number of throws required by Ram and Shyam respectively to obtain six as a result and '$P$' be the probability that $|\alpha -\beta |\le 1,$ then $P$ is equal to $\frac{8}{33}$

Answer 1

Answer: (A), (B), (D)

The correct options are
A Probability that Ram gets six as result on third throw is $\frac{25}{216}$
B Probability that Shyam gets six as result on or after third throw is $\frac{25}{36}$
D Let $\alpha$ and $\beta$ be the number of throws required by Ram and Shyam respectively to obtain six as a result and '$P$' be the probability that $|\alpha -\beta |\le 1,$ then $P$ is equal to $\frac{8}{33}$

$P$(Ram gets six on third throw) $=\frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}$
$P$(Shyam gets six on or after third show) $=\frac{25}{36}$

Let $p$ be the probability for event $|\alpha -\beta |\le 1,$
$p=\underset{\text{both takes same number of attempts}}{\underset{}{\frac{1}{6}\times \frac{1}{6}}}+\underset{\text{one of them takes one more attempt than other}}{\underset{}{(\frac{1}{6}\times \frac{5}{6}\times \frac{1}{6})\cdot 2}}+\underset{\text{one of them takes one more attempt than other}}{\underset{}{\frac{5}{6}\times \frac{5}{6}}\times \left(p\right)}p=\frac{8}{33}$