My Solution : Simpler Problem

Let \((Y_1, \dots, Y_5)\) and \((X_1, \dots, X_5)\) be the random variables that represent the winning percentage of the teams in the East and Central Division.

From the problem definition we know that both \(X_i \xleftarrow[]{\$}[0,1]\) and \(Y_i \xleftarrow[]{\$}[0,1]\) for all \(i \in [5]\) i.e. they are sampled uniformly at random from an interval of \([0,1]\).

The problem is essentially asking if I draw all 10 of these random samples what is the probability that the final ordering has \(X\)’s in the top 5.

There are 10! factorial possible ranking strategies for and \(5! \times 5!\) ways to arrange the different X’s and Y’s in the first 5 and last 5 slots.

Thus the first answer is simply \(\frac{5!\times5!}{10!} = \frac{1}{252}\).

A monte carlo simulation of drawing process confirms the above result