Sample space corresponding to rolling a die

An experiment consists of rolling a die until 3 appears, determine how many elements of the sample space correspond to the event that the 3 appears no later than the k-th roll of the die.

5^0+5^1+5^2+ ... + 5^{k-1} = (5^k-1)/4

I'm wondering is there any other ways to do this question? Maybe involve complements somehow?

thanks

Re: Sample space corresponding to rolling a die

Quote:

Originally Posted by

**usagi_killer** An experiment consists of rolling a die until 3 appears, determine how many elements of the sample space correspond to the event that the 3 appears no later than the k-th roll of the die.

5^0+5^1+5^2+ ... + 5^{k-1} = (5^k-1)/4

I'm wondering is there any other ways to do this question? Maybe involve complements somehow?

I don't think there is a better way.

If $\displaystyle X$ denotes roll on which the first three appears then $\displaystyle P(X = k) = {\left( {\frac{5}{6}} \right)^{k - 1}}\left( {\frac{1}{6}} \right)$.