How hard is it to beat another team 3 times in the same season?

With the Steelers set to play the Ravens in the AFC title game, after Pittsburgh won both of their meetings in the regular season, the old adage about how hard it is to beat someone 3 times in the same season appears to be all over the place. You might think that in this day and age of easily retrievable stats and history, professional journalists might look for the actual facts before pontificating about them, but...nope.

So I did just a bit of Googling, and turned up the answer here: since the AFL-NFL merger in 1970, teams in Pittsburgh's current situation are 11-7 in their playoff matches, against teams they'd beaten twice in the regular season. So the actual facts could probably be interpreted one of two ways: there's a slight advantage to the team that won the first two games, or, because the all-time record is pretty close to even, there isn't much of an edge either way.

In any event, the record certainly shows that it's not unusually hard to beat someone three times in the same season.

If the odds are even, then I see it like this if we are only focused on 2 teams.

Game 1- the odds are 100% one of the 2 teams will win (ignoring ties)
Game 2- the odds are 1 in 2 that the winner of game 1 will win again
Game 3- the odds are 1 in 2 that the winner of 1 and 2 will win again.

So, the odds of 1 of the 2 teams sweeping the 3 game season series is 1 in 4. So it should happen quite often league wide.

Ravens are tired...I see them losing, unless the steelers defense chokes or they have a big play which wasn't a TD.

True enough, but the widespread misinformation on this is more a question of conditional probability. The only time anyone talks about this is before a playoff game where (a) the teams have already played each other twice that season (i.e. they're in the same division) and (b) one team has won both of those games. As you say in your assumption for Game 3, it really doesn't matter who won the first two games, and indeed the historical results are fairly close to a 50-50 proposition for the result of the third game.

Now if we knew before this season that Pittsburgh and Baltimore would be playing each other 3 times--which, of course, we couldn't know without a crystal ball telling us the playoff results up to this point--then the odds of Pittsburgh sweeping that series would indeed be more like 1 in 4. That's perfectly consistent with describing it as something that's "hard to do." But those odds include the cases where Baltimore would have beaten them in the regular season, which is not included now, because that's already in the past.

I took the condition after the 1st game because I was taking the odds of either team sweeping. If we look at it before the season, then the odds of Pittsburgh winning all 3 will be 1 in 8 (assuming as you point out that they were guaranteed 3 match-ups). 1/2 x 1/2 x 1/2. Same odds as Shaq hitting 3 straight free throws.

Reminder that team that was beaten twice must be good or you wouldn't get a 3rd chance. I this case a team with 2 more P/O victories.
I thought [and a lot of writers] that Ravens jumped the shark by quitting on Dilfer after he won the Superbowl. He was the glue that held that team together.

When you're talking about playing the same team 3 times in the same season, it means playoffs. So, by definition, the Lions were excluded from that statement.