An experiment consists of rolling die until a 2 appears. How many elements of the sample space correspond to the event that 2 appears on the kth roll of the die? How many element of the sample space correspond to the event that 2 appears not later than the kth roll of the die?

Explanation

In a through of a die there is 6 sample points.
(i) If 2 appears on the kth roll of the die.
So, first (k-1) roll have 5 outcomes each and kth roll results 2 i.e., 1 outcome.
`therefore" Number of element of sample space correspond to the event that 2 appears on the kth roll of the die" =5^(k-1)`
(ii) If we consider that 2 appears later than kth roll of the die. then it is possible that 2 comes in first throw i.e., 1 octcome.
If 2 does not appear in first throw, then outcomes will be 5 and 2 comes in second throw i.i., 1 outcome, possible outcome `= 5xx1=5`
Similarly, if 2 does not appear in second throw and appears in third throw.
`therefore" ""Possible outcomes"=5xx5xx1`
`"Given"" ""Series"=1+5+5xx5+5xx5xx5+...+5^(k-1)`
`=1+5+5^(2)+5^(3)+...+5^(k-1)`
`=(1(5^(k-1)))/(5-1)=(5^(k-1))/(4)`