# If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?

Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two consecutive selected seats as a single pair.)

For example, if the selected seats are all consecutive (eg 1-25), then we have 24 consecutive pairs (eg 1&2, 2&3, 3&4, …, 24&25). The probability of this happening is 75/($_{100}C_{25}$). So this contributes $24\cdot 75/(_{100}C_{25}$) to the expected number of consecutive pairs.

Motivation: I teach. Near the end of an exam, when most of the students have left, I notice that there are still many pairs of students next to each other. I want to know if the number that remain should be expected or not.

If you’re just interested in the expectation, you can use the fact that expectation is additive to compute

• The expected number of consecutive integers among $\{1,2\}$, plus
• The expected number of consecutive integers among $\{2,3\}$, plus
• ….
• plus the expected number of consecutive integers among $\{99,100\}$.

Each of these 99 expectations is simply the probability that $n$ and $n+1$ are both chosen, which is $\frac{25}{100}\frac{24}{99}$.

So the expected number of pairs is $99\frac{25}{100}\frac{24}{99} = 6$.