Probability problem

[hungryhungarian]

Five people play a game where each player has a chance of ½ for winning any match (for example, a coin toss). All of the five players play against all other four. What is the probability that each player wins exactly two games?

[dgreenb801]

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I think the answer is . Call the teams 1,2,3,4, and 5. First consider the games 1-2, 1-3, 1-4, 1-5.
We can assume 1 beats 2 and 3 and loses to 4 and 5. There is a chance 1 will lose exactly two games. Now consider the game 2-3. We can assume 2 beats 3. Now, the probability 2 will lose to exactly one of either 4 or 5 is , we can assume 2 beats 4, loses to 5. But then all of the rest of the games are determined. 3 and 4 must beat 5, as 5 has already one two games, and then 3 must beat 4. There is a probability these final three games will occur like this. The final answer is .