An experiment consists of rolling die until a 2 appears. (i)How many elements of the sample space correspond to the event that 2 appears on the `k^(th)` roll of the die? (ii)How many element of the sample space correspond to the event that 2 appears not later than the `k^(th)` roll of the die?

Explanation

Number of outcomes when die is thrown is ‘6’.

(i) If `2` appears on the `A^(th)` roll of the die.
So, first `(k -1`) roll have `5 `
outcomes

So,

Number of outcomes = `5^(k-1)`

(ii) If we consider that `2` appears not later than `K^(th)` roll of the die,
then `2` comes before `k^(th)`roll.

If `2` appears in first roll, number of ways `= 1`

If `2` appears in second roll,
number of ways `= 5 xx 1` (as first roll does not result in `2`)
Similarly if `2` appears in `(k – l)^(th)` roll,
number of ways `= [5xx5xx5… (k- 1) ` times`] xx 1 = 5^(k-1)`

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