A soccer trainer is making a team, he can choose 11 out of 22 players. The trainer has to take into consideration the players preferred position:
2 players want to be keeper. 8 players want to play defense. 8 players wants to play middle field. 4 players want to play attack.
The team shall consist of 1 keeper, 4 defensive players, 4 middle field players and 2 attackers. How many different teams can the coach create?
I found: 2C1 * 8C4 * 8C4 * 4C2 = 58800.
Now the follow up question is what confuses me:
John and Peter wants to play middle-field.
Given that the coach takes into consideration of the player preferences (like above), what are the odds that:
John makes the team as middle-field player
Peter makes the team as middle-field player
Peter making the team as a middle-field player if we know that John made the team as a middle-field player.
I understand that I have to apply the conditional probability theorem, but I am unsure as to how.
Any help is much appreciated.
The number of ways to not choose Peter is $7\choose 4$. The total number of ways to choose a team altogether is $8\choose 4$. Therefore the answer is the ratio netween these two, which happens to be the much expected $1/2$.
As for Question 3, given that we have one player already, there are only $3$ to choose from $7$ now, and the updated figures are $$\frac{6\choose 3}{7\choose 3}=\frac{4}{7}$$
therefore the probability that Peter makes it is $3/7$.