Two players A and B throw a die alternately for a prize of Rs 11, which is to be

Question

Two players A and B throw a die alternately for a prize of Rs 11, which is to be won by a player who first throws a six. If A starts the game, their respective expectations are

Options:

A . Rs 6; Rs 5

B . Rs 7; Rs 4

C . Rs 5.50; Rs 5.50

D . Rs 5.75; Rs 5.25

Answer: Option A
:
A
Probability through a six =

\frac{1}{6}

P (A) = \frac{1}{6}, P (¯ A) = \frac{5}{6}, P (B) = \frac{1}{6}, P (¯ B) = \frac{5}{6}

Probability of A winning

= P (A) + P (¯ A) P (¯ B) P (A) + P (¯ A) P (¯ B) P (¯ A) P (¯ B) P (A) + . . .

= \frac{1}{6} + \frac{5}{6} \times \frac{5}{6} \times \frac{1}{6} + \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{1}{6} + . . .

= \frac{\frac{1}{6}}{1 - \frac{25}{36}} = \frac{6}{11}

Probability of B winning

= 1 - \frac{6}{11} = \frac{5}{11}

∴

Expectations of A and B are

\frac{6}{11} \times 11 =

Rs 6 and

\frac{5}{11} \times 11

=Rs 5

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