Probability Puzzle: Exactly one of two specific balls among $N$ balls in $n$ draws. We have an urn with $N$ different balls of all colours, including one white and one red ball. We draw $n$ times without replacement.
After $n$ iterations, what is the probability that among the sampled balls is either the red or the white ball but not both?
 A: If $n\ge N$ then the probability is $0$ since you have drawn all the balls. Otherwise, if $n<N$ you can use the Hypergeometric distribution. Specifically, the number of successes $X$ (where as a success we denote to draw the red or the white ball), is a hypergeometric random variable (since drawing is performed without replacement) with parameters $N$ (population size), $2$ (number of successes in the population) and $n$ (sample size). 
You want to calculate the probability $P(X=1)$ which - by using the pmf of the hypergeometric distribution - is equal to $$P(X=1)=\dfrac{\dbinom{2}{1}\dbinom{N-2}{n-1}}{\dbinom{N}{n}}=2\dfrac{n(N-n)}{N(N-1)}=2\dfrac{n}{N-1}\left(1-\dfrac{n}{N}\right)$$
A: The probability that the red is in our sample is $\frac{n}{N}$. 
The probability that the white is in our sample given that we have the red is $\frac{n-1}{N-1}$. 
So the probability that we have the red but not the white is
$$\frac{n}{N}\left(1-\frac{n-1}{N-1}\right).\tag{1}$$
For our required probability, double the expression in (1).
