The Theater Row Brain Teaser-why the probability is constant? for the classic brainteaser, The Theater Row:
Eight elegible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?
The explanation of the question says we just multiply the probability of getting the first mixed gender pair multiply the number of pairs in the 15-seat row. What I don't get is that, I think the probability of getting mixed gender pairs is not constant. For example, the prob of getting the first mixed gender is 8/15. However, shouldn't the probability of getting the 2nd, 3rd, ..,ith mixed gender pair depend on the number of such mixed pair before that one? Why we can just assume the probability is constant?

This boils down to the linearity of expectation. Let $X_{i,i+1}$ be a random variable that is $1$ if the occupants of seats $i$ and $i+1$ consist of one bachelor and one model (or vice versa). You are interested in
$$E[X_{1,2} + X_{2,3} + \cdots + X_{14, 15}].$$
By linearity of expectation, this equals
$$E[X_{1,2}] + E[X_{2,3}] + \cdots + E[X_{14, 15}].$$
Thus you can compute each term separately. This is extremely convenient, since to compute $E[X_{2,3}]$ (which ends up being "probability that seats $2$ and $3$ are occupied by one bachelor and one model (or vice versa)"), you can ignore what happens outside of the 2nd and 3rd seats. It turns out $E[X_{1,2}] = E[X_{2,3}] = \cdots = E[X_{14, 15}]$; when focusing on a particular pair of adjacent seats, it doesn't matter which pair you are focusing on.
