An urn has n white and m black balls that are removed one at a time in a randomly chosen order. Find the expected number of instances in which a white ball is immediately followed by a black one.
How do I approach such questions? I am clueless about where to even begin!
Best Answer
For $k=1,\ldots,m+n-1$ let $X_k$ be $1$ if the $k$-th ball is white and the $(k+1)$-st ball is black, and $0$ otherwise. There are $\binom{m+n}m$ possible sequences of white and black balls, all of which are equally likely. If we specify that the $k$-th and $(k+1)$-st balls are white and black, respectively, there are $\binom{m+n-2}{m-1}$ possibilities for the remainder of the sequence; why? This means that
$$\Bbb E[X_k]=\frac{\binom{m+n-2}{m-1}}{\binom{m+n}m}=\frac{(m+n-2)!\,m!\,n!}{(m+n)!\,(m-1)!\,(n-1)!}=\frac{mn}{(m+n)(m+n-1)}$$
for $k=1,\ldots,m+n-1$. Now express the number of white balls that are immediately followed by a black ball in terms of the random variables $X_k$, and use linearity of expectation to get the desired result.