TITLE: Probability with unknown variables
QUESTION [7 upvotes]: An urn contains $10$ red marbles and $10$ black marbles while a second urn contains $25$ red marbles and an unknown number of black marbles. A random marble will be selected from each urn and the probability that both marbles are the same will be determined. A hint was given by the teacher: the probability does NOT depend on the number of unknown marbles. Verify that this is the case.
Let's call $N$ the unknown number of marbles.
I wrote out all possible ways to select a marble from each urn, selecting a red marble from both urn $1$ and urn $2$, and selecting a black marble from urn $1$ and $2$ and this is what I got:

Number of ways to select a marble from each urn: $ \binom{20}{1}\binom{25+N}{1}$
Number of ways to select $1$ red marble from both urn $1$ and urn $2$: $\binom{10}{1}\binom{25}{ 1}$
Number of ways to select $1$ black marble from urn $1$ and urn $2$: $\binom{10}{1}\binom{N}{ 1}$

And this is what I got as my final equation to finding out the probability of selecting the same color marble from each urn: $\dfrac{\binom{10}{1}\binom{25}{ 1}+\binom{10}{1}\binom{N}{ 1}}{\binom{20}{1}\binom{25+N}{1}} $
I am confused on how the probability doesn't depend on the unknown number of black marbles in urn $2$? Any help would be much appreciated, thank you so much!
PS: I also searched through stack exchange for a problem similar to this and I couldn't find one. If this question was asked already, then I apologize!

REPLY [5 votes]: Hint:
$\dfrac{\binom{10}{1}\binom{25}{ 1}+\binom{10}{1}\binom{N}{ 1}}{\binom{20}{1}\binom{25+N}{1}} =\frac{\binom{10}{1}\cdot\left[\binom{25}{ 1}+\binom{N}{ 1} \right]}{20\cdot(25+N)}=\frac{10\cdot (25+N)}{20\cdot (25+N)}$