Shortly I posted a similar question to this (here: Probability for "drawing balls from urn"). Actually, the following question was included in this topic as well::
In an urn there are $N$ balls, of which $N-2$ are red and the remaining are blue. Person $A$ draws $k$ balls, so that the first $k-1$ are red and the $k$ ball is blue. Then Person $B$ draws $m$ balls $\dots$:
What's the probability for Person $A$ to draw the first blue ball after drawing $k$ red ones? Meant is, that the drawing stops as soon as the blue ball has been drawn. So the first blue ball must be the last one for Person $A$
My ideas - following the answer from Jimmy R. (= direct approach):
- $P(X=1) = \dfrac{2}{N}$
- $P(X=2) = \dfrac{N-2}{N} \cdot \dfrac{2}{N-1}$
$\dots$
Following this pattern I finally get
$$P(X=k) = \frac{\dbinom{N-2}{k}}{\dbinom{N}{k}} \cdot \frac{2}{N-k}$$
In fact, this solution differs only slightly from doing it via "Hypergeometric Distribution", which I did first. However, if I got it right then the Hypergeometric Distribution should be the wrong thing in this case. What remains is an explanation for Hypergeometric Distribution being not what I want, but here I stuck:
I'm citing wiki ("Hypergeometric distribution"): In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes, wherein each draw is either a success or a failure.
Let's see what I have here:
- $k$ successes in $n$ draws? Yes, if I see a success as "drawing blue ball" so I have $k=1$ success
- without replacement? Yes, since I won't put any ball back.
- a finite population of Size $N$? Yes, I have $N-2$ red ones and $2$ blue ones.
- exactly $K$ successes? Yes, in my case $K = 2$. I have two blue ones, so two successes.
So why won't this help me?
EDIT: Possible answer: It won't help, since I'm not interested in "$1$ success among $k$ draws" but in "$1$ success at the END of $k$ draws"?