The NFL is the professional American football league.  Each year, every team plays 16 games against 16 opponents.  The goal?  Make the post-season.  The rules for getting in are a little complex.

Each team belongs to a division which is 4 teams.  There are 8 such divisions, so a total of 32 teams.  Half the divisions belong to one conference called the AFC.  The others belong to the NFC.

The goal for each team is to win their division by having the best win-loss record.  Beyond that, if they are the top 2 teams in their conference (which has 16 teams), they get a first-round bye.  This bye keeps teams with good records playing hard until the end to get that bye.  Not all divisions are equally strong, so sometimes you can sneak into the playoffs because your division has bad teams and get into the playoffs when another team places second in a much stronger division and may even have a better record than you (though usually just barely) and you get in, and they don’t.

The scenarios get quite complex at the end, as some statistician tells the teams how many games they must win to get in, and which other teams must lose.  It’s complex enough, most people don’t even bother trying to do the computation.

However, in a round robin, with only 3 rounds, you can generally do the math by playing out the scenarios.  It just takes some patience.

To understand how to do the math, let’s start easy.  With 3 matches, you can only have 4 possible results: 3-0, 2-1, 1-2, and 0-2 with each successive record worse than the previous.

So, 3-0 beats 2-1.

The top two players make the semifinals in each group (there are two groups of 4).  Because of this, even a 2-0 lead or a 0-2 deficit is no guarantee you can’t make the semis.

If several players have the same record, say, three players finish 2-1 or three players finish 1-2, then the next tiebreak to decide who makes the semis is the percentage of sets won.  If there’s still a tie after that, they break it by percentage of games won.  In 2009, this scenario happened.  Federer, Murray, del Potro, and Verdasco were in the same group.  All 6 matches went three sets and Verdasco lost all 3 of his matches.  So, each of Federer, Murray, and del Potro had won the same number of matches and lost the same number, and won the same number of sets, and lost the same number.  It came down to percentage of games won and del Potro won one more game than Murray (and both lost the same number of games) and Delpo was in and Murray was out.

OK, back to 2010.  After the second match of round robin play in Group A, Rafael Nadal had a 2-0 lead while Andy Roddick had a 0-2 deficit.  Yet, there were scenarios that Roddick could make the semifinals as the number 2 guy and other scenarios that Rafa could miss the semifinals.  Admittedly, there were no scenarios when Roddick makes it in, but Rafa doesn’t.  Rafa already had 2 wins and Roddick could, at best, only have 1 win.

How does Andy Roddick make it in?  With Rafa at 2-0, Berdych and Djokovic at 1-1, then, the only way Roddick can make it in with 1 win is for both Berdych and Djokovic to lose.  That way, both of them have 1 win each too.

Since Berdych would play Rafa, then Rafa, at 2-0 would have to win, and he’d be 3-0.  Roddick would have to beat Djokovic.  If that happens, Rafa is 3-0, and Berdych, Djokovic, and Roddick are 1-2.  At this point, percentage of sets on is the second tiebreak to determine who gets into the semis.  (Rafa would already be in by virtue of sweeping the group).

Here’s where Roddick has his one and only advantage.

He took a set off of Rafa.

This is critical.  Both Berdych and Djokovic lost their matches in straight sets.  If Roddick beats Djokovic in straight sets and Rafa beats Berdych in straight sets, both somewhat realistic (though Roddick beating Djokovic is the harder of the two), then Roddick would be the other semifinalist because of that set he won against Rafa.

In fact, Roddick can’t afford to lose a set to Djokovic.  Djokovic has lost 2 sets already (to Rafa), but Roddick has lost 4 sets (to Rafa and Berydch).  If Djokovic wins a set in a loss, he’ll have won the same number of sets as Roddick (3 sets), but Djokovic will have lost 4 sets while Roddick, in giving up a set, will have lost 5, so Roddick’s win percentage would be 3/8 while Djokovic will be 3/7.  If Roddick wins in straights, the win percentage would be 3/7 while Djokovic would be 2/6.

There’s also no way for both Roddick and Berdych to make it together.  Either Berdych wins over Rafa, in which case both of them have 2 wins each, and they both go to the semis, or Berdych loses and Roddick wins, and then Rafa makes it, and only one of Djokovic, Berdych, and Roddick make it.

For a similar reason, Roddick and Djokovic can’t both make the semifinals.  The reason?  Berdych must lose if Roddick is to have a chance.  If Berdych wins, he and Rafa have 2 wins, and Roddick is definitely out (he can only gain 1 win).  That means, Rafa has to win, and if he does that, Rafa is definitely one of the semifinalists at 3-0.

How can Rafa not make it?  Basically, Rafa has to lose to get to 2-1 and both Djokovic and Berdych, who are 1-1, have to win.  In this case, Rafa’s one set loss to Roddick may come to haunt him.  If Djokovic wins over Roddick in straight sets and Berdych beats Rafa in straight sets, both Djokovic and Berdych will have won 4 sets out of 6 while Nadal will have won 4 sets out of 7 and Nadal would be heading home.

So, with a little math (or maths, as the British like to say), you can figure it out!