Question

Two persons A and B are throwing an unbiased six faced die alternatively, with the condition
that the person who throws 3 first wins the game. If A starts the game, the probabilities
of A and B to win the same are respectively.

• A. $\frac{6}{11},\frac{5}{11}$
• B. $\frac{5}{11},\frac{6}{11}$
• C. $\frac{8}{11},\frac{3}{11}$
• D. $\frac{3}{11},\frac{8}{11}$

Answer 1

The correct option is A $\frac{6}{11},\frac{5}{11}$

A and B throws the dice simultaneously

Probability of getting 3=1/6

A will win the game if A throws 3 before B

Probability of A winning the game P(A)=1/6+5/6*5/6*1/6+5/6*5/6*5/6*5/6*1/6+....

its a sum of infinite GP P(A)=1/6(1+5/6*5/6+5/6*5/6*5/6*5/6+.....)

=1/6*(1/(1-25/36)

=1/6*(36/11)=6/11

B will win the game if B throws 3 before A

Probability of B winning the game

P(B)=5/6*1/6+5/6*5/6*5/6*1/6+....

its a sum of infinite GP P(A)=(5/6)*(1/6)*(1+5/6*5/6+5/6*5/6*5/6*5/6+.....)

=5/36*(1/(1-25/36)

=5/36*(36/11)=5/11