A fair die is rolled. The probability that the first time 1 occurs at the screen throw, is

$\frac{5}{11}$

Let $A-i$ denote the event of getting 1, first time, in $i^{th}$ throw.

Clearly, p = Probability of getting in a throw $=\frac{1}{6}$

q = Probability of not getting 1 in a throw $=\frac{5}{6}$

Required probability $=P(A_2 ∪ A_4 ∪ A_6 ∪ ...)$

$= P(A_2)+P(A_4)+P(A_6)+P(A_8)+...$

$qp +q^3p+q^5p+q^7p+....$

$=\frac{qp}{1-q^2}=\frac{5/36}{1-\frac{25}{36}}=\frac{5}{11}$