For the Red Sox to not at least tie for a playoff spot, a lot of things would have to go exactly right (or wrong, depending on your perspective) at this point. 3 teams would have to overtake Boston to knock the Red Sox out of both the division title and both wild card spots. As it turns out, only 3 teams in the American League that can still overtake the Red Sox in the race for the 3 playoff spots available to the Red Sox: the Rays, the Indians, and the Rangers. Since Baltimore can only tie at this point by winning all their remaining games, and since doing so would knock the Rays out of contention by giving the Rays at least 4 losses when they can only afford 3 to catch the Red Sox, the only scenario that can work is the Rays beating Baltimore.
Because Cleveland has 69 losses against Boston’s 92 wins, they must win all their 12 remaining games, and the Red Sox must lose all their 11 remaining games, for Cleveland to overtake them.
Tampa Bay, with 67 losses, can afford to lose no more than 2 games and still overtake the Sox; Texas, with 68 losses, can afford to lose no more than 1 game. These teams are scheduled to play 3 more games against each other, however, so between them they must lose at least 3 more games. Therefore, for both to overtake the Sox, the Rangers must beat the Rays 2 games out of 3, and both teams must win all their other remaining games (10 each).
So between the 4 teams, Boston, Cleveland, Tampa Bay, and Texas, 43 scheduled games must go a certain way, while the other 3 have a little more flexibility, with 3 of the 8 possible outcomes working to bump out the Red Sox.
If we assume each game has a 50-50 chance of going to one team or the other, that makes (3/8)*(1/2 to the 43rd power) as the chances of this happening: that’s less than one in 23 trillion.
However, the chances of the Red Sox losing any one of their games has to be less than 50% right now, and with Texas scuffling through a 7-game losing streak right now, I’d say the odds are even slimmer.
Yeah, it’s not yet a mathematical gaurantee, but come on, do you really need better odds than that?
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