From Nate Silver:
“The following is not mathematically rigorous, since the events of yesterday evening were contingent upon one another in various ways. But just for fun, let’s put all of them together in sequence:
• The Red Sox had just a 0.3 percent chance of failing to make the playoffs on Sept. 3.
• The Rays had just a 0.3 percent chance of coming back after trailing 7-0 with two innings to play.
• The Red Sox had only about a 2 percent chance of losing their game against Baltimore, when the Orioles were down to their last strike.
• The Rays had about a 2 percent chance of winning in the bottom of the 9th, with Johnson also down to his last strike.
"Multiply those four probabilities together, and you get a combined probability of about one chance in 278 million of all these events coming together in quite this way.”
Nate speculates that the outcomes might have been correlated rather than independent, although this might have to depend on statistically hard-to-support claims about systematically "clutch" and "un-clutch" behavior. But he also offers an amusing stick figure illustration, entitled "All Sports Commentary," in which the first person says: "A weighted random number generator just produced a new batch of numbers."
The second one replies: "Let's use them to build narratives."