# An experiment consists of rolling a die until a 2 appears.How many elements of the sample space correspond to the event that the 2 appears not later than the $k^{th}$ roll of the die?[Hint (a) first (k-1) rolls have 5 outcomes each and $k^{th}$ rolls should result in 1 outcomes.(b) $1+5+5^2+.......5^{k-1}$]

$\begin{array}{1 1}(A)\;\large\frac{5^{k-1}}{4}\\(B)\;5^{k+1}\\(C)\;5^k\\(D)\;\text{None of these}\end{array}$

The sample space consists of the collection of finite sequences $(x_1,x_2........x_k)$ such that the $k^{th}$ term in each sequence is 2 and all other terms are one of 1,3,4,5,6
$\therefore$ First (k-1) rolls have 5 outcomes each and $k^{th}$ rolls should result in 1 outcomes.
$\therefore$ Event that the 2 appears on the $k^{th}$ roll of the die =$5^{k-1}$ elements.
Similarly for the event that the 2 appears not later than $k^{th}$ roll of the die =$5^0+5^1+5^2+........5^{k-1}$
$\Rightarrow \large\frac{5^{k-1}}{4}$
Hence (A) is the correct answer.