I have this problem:
Under the corks of a certain brand of beer there are six different types of symbols. Assuming that in the market the beers were evenly distributed. If you are challenged to drink beers until you have the six symbols, what is the expected number of beers you will drink?
I tried to define a random variable X which represents the number of the beers that were distributed (X ~ U{x0, ..., xn}), but then I don't really know how to continue.
Is there an idea?
Use linearity of expectation. We can write the total time as $$T_1+T_2+\ldots T_6$$ where $T_1$ is the time to get the first one, $T_2$ to the second (after the first), and so on. $T_1=1,$ so $E(T_1)=1$ of course. Then to get the second all you have to do is not get the first one, so $E(T_2) = 6/5.$ Then to get the third one, you only have a $2/3$ probability of seeing something new, rather than $5/6,$ and so on.