We choose randomly a subset $\mathit S$ of size 25 from the set $\{ 1,2,...,100\}$ , what is the expected value of number of Consecutive numbers in $\mathit S$?
Consecutive numbers in $\mathit S$: is a pair $\{ i,i+1\}$ where i$\in$$\mathit S$ and i+1$\in$$\mathit S$.
Linearity of expectation is the key. There are 99 consecutive pairs, each one having a $\frac{\binom{98}{23}}{\binom{100}{25}}=\frac{25×24}{100×99}$ chance of being selected. Thus the expected number of consecutive pairs is $\frac{25×24×99}{100×99}=6$.