Question

10 kids are randomly grouped into an A team with five kids and a B team with five kids. Each grouping is equally likely. There are three kids in the group, Alex and his two best friends, Jose and Carl. Define the events J and C as:

• J: Alex ends up on the same team as Jose
• C: Alex ends up on the same team as Carl

Are the events J and C independent? Prove your answer by showing that one of the conditions for independence is either true or false.

Answer 1

Solution:

The size of the sample space is C(10, 5). The number of ways to select the teams so that Alex and Jose are both on the A team is C(8, 3) because once Alex and Jose are placed on the A team, there are C(8, 3) ways to select the remaining three kids for the A team. A similar argument holds for Alex and Jose to be placed on the B team together. Therefore |J| = 2 C(8, 3) and p(J), the probability that Jose and Alex are on either team together, is 2 C(8, 3)/C(10, 5) = 4/9.

p(J|C) = |J ∩ C|/|C|. We can use the same reasoning above to conclude that |C| = 2 C(8, 3). J ∩ C is the event that Alex is on the same team with both Carl and Jose. The number of ways for Alex, Jose and Carl to be all on the same team is 2 C(7, 2). (There are two possible teams. Then once the three kids are placed on one of the two teams, there are C(7, 2) ways to select the two other kids for that team.) p(J|C) = |J ∩ C|/|C| which is 2 C(7, 2) / 2 C(8, 3) = 3/8.

p(J) = 4/9 ≠ 3/8 = p(J|C). The events J and C are not independent.