# An interesting probability calculation problem

This is my first question in this site. I'll be glad if you respond to this question. I am working with an interesting probability calculation problem. In my problem there are ratings for 32 chess players. They are divided into two groups to form two chess teams in such a way that each team contains all types of players. However, each team should contain only 4 players and the whole design for selecting the players for the two teams has been made according to the following diagram:

32 players are randomly divided into two groups first, each containing 16 players. Each of these 16 players are further randomized into 4 groups each containing 4 members. Then within each group, the players are arranged in ascending order of their ratings. Then the players with lowest rating in the 1st group, 2nd lowest in the 2nd group, 3rd lowest in the 3rd group and highest in the 4th group are selected to from a team.

Given this design, what is the probability that player i, (i=1,...,32) will be assigned to Chess Team-1?

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Since there's no difference between player i's chance of being on Team 1 and player j's chance of being on Team 1, the probability is 1/8 for everybody--1/2 probability of being in the pool of 16 that forms Team 1, and 1/4 of being selected for the team.

Here's a function written in R to simulate the selection of Team 1:

simulate.team <- function() {
# We assume the players are ranked best to worst, 1-32
group1 <- sample(1:32, 16) # Select half the players at random for Group 1
subgroup1 <- group1[1:4] # Form subgroups. Since group1 is already in a
subgroup2 <- group1[5:8] #   random order, we can just take the first four,
subgroup3 <- group1[9:12]#   then the second four, etc.
subgroup4 <- group1[13:16]
team1 <- c(sort(subgroup1,decreasing=TRUE)[1], # select the players that
sort(subgroup2,decreasing=TRUE)[2], #   will be on Team 1.
sort(subgroup3,decreasing=TRUE)[3],
sort(subgroup4,decreasing=TRUE)[4])
return(team1)
}


When we simulate it 100,000 times and plot a histogram of how often each player ends up on the team, we get the following:

As you can see, the probability is the same for every player (up to simulation error).

If you are still unconvinced, I encourage you to write your own code to simulate the process.

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16 people are further divided into 4 groups and the order of the rating should matter. –  Blain Waan Jan 17 '13 at 18:30
I know. The order of the rating doesn't matter: no matter which group of 4 people the player is placed into or where the player ranks among those four people, he has a 1/4 chance of being selected for the team. This is because the four groups of four are chosen randomly. If, for example, the top four were in one group, the next four in the second group, etc. then obviously the probabilities would not be all equal. –  Jonathan Christensen Jan 17 '13 at 18:37
@BlainWaan I have updated my post to include a simulation of the process, which agrees perfectly with my reasoning. Can you explain why you think I'm wrong? –  Jonathan Christensen Jan 23 '13 at 15:10