An unfair die is rolled until both an even number and odd number have appeared on top. The probability that the roll shows $n$ is proportional to $n$, where $n$ = 1,2,3,4,5,6.
What is the expected number of rolls, and what is the probability that the last roll shows even?
I believe the answer to the latter is $12/21$, but I am not sure. Additionally, I am unsure of how to compute the expectation value for the number of rolls until both even and odd have shown.