Combinatorics: Dividing 10 people into 2 groups of 5 simultaneously This is relevant to a video game, where 10 people are found and divided into 2 teams of 5 randomly. Suppose the grouping happens in one go, by that I mean teams are not assigned to participants one at a time.
You and another person, T are among the 10 people. I understand that if given you belong to a certain team (A), then the chances of T being in your team is 4/9. 
However, if the grouping of 10 people happen simultaneously, isn't the probability of you two ending up on the same team 1/2?'
Thanks you.
 A: People are not equally likely to be on the same team than on different teams, even if they are all assigned their team simultaneously. This is because there are more possible teams given that You and T are on different teams than they are on the same team. This might seem unintuitive, but will make more sense after some analysis.
Let's say You are on Team A and T is on Team B. Team A has four spots other than You and there are eight other players, meaning that there are ${8 \choose 4}$ possible Team As. Now, given a Team A, we always know that the Team B is going to be the other five people, so we don't need to account for the other possibilities on Team A.
Now, let's say You and T are on Team A. Team A has three spots other than You and T and there are eight other players, meaning that there are ${8 \choose 3}$ possible Team As. Again, given a Team A, we always know that the Team B is going to be the other five people, so we don't need to account for the other possibilities on Team A.
Now, we can calculate the probability by dividing the number of possibilities with You and T both on the same team by the total number of possibilities:
$$\frac{{8 \choose 3}}{{8 \choose 4}+{8 \choose 3}}=\frac 4 9$$
Just to be clear, we are talking about possibilities of the final teams. This has nothing to do with assigning You and T's team first and then figuring out everyone else's team. It does not matter if we assign the players to Team A first and then the players to Team B or we do it in the other order or we do it simultaneously. The order of assignment simply does not matter. If we look at possibilities, then we get rid of order entirely and don't have to worry about how the assignment is done because we only care about the final teams. This is a common way I like to think about probability problems because it makes certain problems like these a lot simpler.
