Dice rolling, even results

Question

When rolling a dice until you get 6, the expected number of rolls is 6 (geomteric dist). How do I get the expected value of number of even results? Last result would be 6 which is even. How do I account for the rest?

Thanks.

Answer 1

You score $1$ point for each roll that comes out even, $0$ for each that comes out odd, stopping when you get a $6$.

All rolls save the last are drawn uniformly and independenly at random from $\{1,2,3,4,5\}$. You expect $6-1=5$ such rolls, scoring an average of $5/2=0.4$ points on each, for a total of $2$ points - to which you should add your final point for scoring a $6$. A grand total of $3$ expected even rolls.

Another way to approach the problem is to imagine rolling and rolling and rolling, and splitting your gigantic sequence of rolls into subsequences - ending the current subsequence every time you roll a $6$. Your subsequences are going to be in expectation (as you correctly wrote) one sixth the number of rolls, while your even rolls are going to be half the number of rolls, so you'll get on average $6/2=3$ even rolls per subsequence.