Question

A and B throw with one die for a stake of Rs. 11 which is to be won by the player who first throws a 6. If A has the first throw, what are their respective expectations?

• A. Rs. 7, Rs. 4
• B. Rs. 6, Rs. 5
• C. Rs. 4, Rs. 7
• D. Rs. 5, Rs. 6

Hint:

When turns alternate, set up an equation for \(P_A\) using first-round win probability plus probability of restarting the game.

Answer 1

The Correct Option is B

Probability that A wins on his first throw: \[ P(\text{A wins immediately}) = \frac{1}{6} \] If both A and B fail in the first round: \[ P(\text{no win in first round}) = \frac{5}{6} \times \frac{5}{6} = \frac{25}{36} \] After such a round, the game restarts with A’s turn, so: \[ P_A = \frac{1}{6} + \frac{25}{36} \cdot P_A \] Solving: \[ P_A \left(1 - \frac{25}{36}\right) = \frac{1}{6} \] \[ P_A \cdot \frac{11}{36} = \frac{1}{6} \] \[ P_A = \frac{6}{11} \] Therefore: \[ P_B = 1 - P_A = \frac{5}{11} \] Expected winnings for each (stake = Rs. 11): - A: \( \frac{6}{11} \times 11 = Rs. 6 \)
- B: \( \frac{5}{11} \times 11 = Rs. 5 \)
\({\text{A: Rs. 6, B: Rs. 5}}\)