# The probability of two spies on the same team, and the probability after switching?

I was watching a TV show, and the setting is pretty simple:

There are 10 players in total, within them there are 2 "spies", players initially form 2 teams of 5 people, and there are 3 rounds of game, where at the end of each round, we can switch a player from team A with a player from team B (or no switches, team members remain unchanged), 2 "spies" win if they are in the same team when the game finishes, otherwise they lose and the rest of the players win

of cause this is a strategy game...spies know their identity and will try to manipulate others

TL;DR here is my question:

one of the player said it is better to maintain the original teams since it is unlikely that 2 spies are in the same team by initial arrangement (random)

well, I got that the initial probability of 2 spies in the same team is $4\over9$ which is indeed less that 50%, but randomly switch a player from team A with another player in team B after round 1 gives me the same probability...so there is no dominant strategy?

here is my calculation: (case 1) 2 spies start being in the same team, after round 1, we randomly switch two players between 2 teams, the probability of 2 spies still in the same team is:$${4\over9}*{3\over5}$$ 3/5 is the probability of non-spy player being selected from the spy team

(case 2) 2 spies start not being in the same team, after round 1, we randomly switch two players between 2 teams, the probability of 2 spies accidentally being in the same team is:$${5\over9}*({1\over5}*{4\over5}*2)$$

1/5*4/5 is the probability of a spy switched with a non-spy player, since 2 spies can end up in either team, (times 2 is required)

sum of these two probabilities gives me again $4\over9$. Is my calculation or the understanding of this game correct?

• The typical person doesn't understand probability very well. Intuitively we think if we change things we "going toward" some state of randomness. You are right, the guy on tv is wrong. It's like people saying "we just had two heads in a row so it's more likely that the next will be a tail". I was playing a game where we had to draw cards. The guy who got the last card said it wasn't fair because he had fewer choices so his chances of avoiding the bad cards got less. In all these cases (yours too) the chances don't change. Commented Nov 30, 2015 at 3:12