Minimum number of wins needed to make the playoffs. Sports meets maths.

You have a 10 man football league.
There are 13 games in the regular season.
Each team plays each other once (9 games) and 4 other random teams twice (4 games).
There are no draws and so every match has a winner and a loser.
The 6 teams with the most wins make the playoffs.

What is the minimum number of wins needed to guarantee a playoff spot?

Assuming every team wins 3 games, what is the minimum number of wins needed.

We have 3 university maths majors working on this and encountering problems with the randomisation. Any clues or answers?

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