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  4. In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six The game stops as soon as either of the players wins. The probability of A winning the game is :

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Q. In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A $ wins the game if he throws a total of $6$ before $B$ throws a total of $7$ and $B$ wins the game if he throws a total of $7 $before $A$ throws a total of six The game stops as soon as either of the players wins. The probability of $A$ winning the game is :

JEE MainJEE Main 2020Probability - Part 2

Solution:

$P (6)=\frac{5}{36}, \quad P (7)=\frac{1}{6}$
$P ( A )= W + FFW + FFFFW +\ldots .$
$=\frac{5}{36}+\left(\frac{31}{36} \times \frac{5}{6}\right) \times \frac{5}{36}+\left(\frac{31}{36} \times \frac{5}{6}\right)^{2} \times \frac{5}{36}+\ldots$
$=\frac{\frac{5}{36}}{1-\frac{155}{216}}=\frac{5}{36} \times \frac{216}{61}=\frac{30}{61}$