Sample space corresponding to rolling a die

• Aug 9th 2012, 05:26 AM
usagi_killer
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
• Aug 9th 2012, 07:00 AM
Plato
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 $X$ denotes roll on which the first three appears then $P(X = k) = {\left( {\frac{5}{6}} \right)^{k - 1}}\left( {\frac{1}{6}} \right)$.