Suppose we have a bag with $3n$ marbles. $2n$ are red, $n$ of them are white. Each turn we pick a marble at random from the bag, look if its a red or white, if its white we return it into the bag, if its red we are done.
What is the expected number of turns we have to take out a stone from the bag in order to get a red stone?
I need to somehow turn this into a series to prove that it is $O$ constant, however I don't know how to approach this. logically it should be $2/3$ which is $O(1)$ but that is not the proof that I am required to do...
Any suggestions are appreciated! T
Thanks!
This can be modeled by a geometric random variable, $X$ where the probability of success (picking a red ball) is $p=2n/3n=2/3$ and failure (drawing a white ball) is $q=1-p=1/3$. The probability of drawing red on the $j$-th trial is $P(X=j)=q^{j-1}p$, for $j\in \mathbb{N}$. Do you know the expectation of such a random variable to conclude? If not, it would be a good exercise to compute it by definition or via PGFs.