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?

If a player is switched at random (i.e., disregarding the element of politics and trickery), then the probability should certainly remain the same. 
Consider shuffling a deck of cards, and suppose you "win" if the ace of spades is on top. Before looking at the top card, you're allowed to make any two cards switch places. Would doing so give you a better chance of winning?
