Baseball Break (1/20 – 2/2)
To play a game of baseball, 44 boys and 22 girls decided to split up randomly into 6 teams of 11 people each. Let \(A_1, A_2, \dots, A_6\) be the number of boys in team 1 through 6, and let \(B_1, B_2, \dots, B_6\) be the number of girls in team 1 through 6. Find the expected value of
\[A_1B_1 + A_2B_2 + \dots + A_6B_6\]
Solution:
Consider a pair \(b_i, g_j\) of a boy and a girl. They contribute 1 to the sum if and only if they are in the same team. By symmetry, we can say that
\[E(A_1B_1 + A_2B_2 + \dots + A_6B_6) = 44 \cdot 22 \cdot \frac{10}{65} = \boxed{\frac{1936}{13}}\]