Question

18 teams, 5 of which are professional, take part in a football tournament. At the beginning of the tournament the 18 teams are randomly divided into two groups (A and B) of 9 teams each. Determine the probabilities of the following events:

a) all professional teams end up in one group

b) 2 of the professional teams are in one group and the other 3 professional teams are in the other group

Answer 1

Let the 5 professional team be considered a one block only.

So total no. of teams now left = 13+1= 14

Now, all these teams can be placed in any of the two groups, except the last team, which will have no option but to join one group.

So 13 teams have two option while last team has only 1 option.

So maximum no. of total combination = 2^13*1

If the teams could join any of the group without any professional/non professional bias, maximum no. of combinations = 2^17*1

Therefore probability = 2^13/2^17= 1/16

2. Let two teams be made one block and place in one group and three teams be made another block and placed in another team.

Now the total no. of places left in both the groups cumulative = 13.

Now the maximum no. of combinations by which the 13 teams can be places in these 13 places = 2^12*1

If the teams could join any of the group without any professional/non professional bias, maximum no. of combinations = 2^17*1

Therefore probability = 2^12/2^17, but since the two blocks of 2 and 3 can also be interchanged hence probability= 2*2^12/2^17= 2^13/2^17= 1/16

Note- Choosing which professional team to place where is immaterial.Here teams are as indifferent as white balls. Nothing