The binomial distribution indicates the probability of Cano going 3-40 (or worse) is 0.00048 - approximately 2084-1 odds.
But that is not the most improbable streak Cano was on in the last month. What I mean is that Cano closed the regular season by going 24-39; that works out to a .615 batting average.
The probability of going 24-39 (or better) is 0.00006 - approximately 16558-1 odds.
In terms of explaining how this could happen, "That's baseball Susan" is all that comes to mind.
To confirm how rare both of these streaks are, I checked my play-by-play database (which goes thru 2011). According to that:
- Cano had never gone 24-39. The best batting average Cano had ever had in 39 ABs was .538 - 21 hits.
- Cano had never gone 3-40. The worst batting average Cano had ever had in 40 ABs was .100- 4 hits.
That's baseball Susan.
Of course, like most stats involving hot and cold streaks, it depends upon how you slice them up. If you combine these 2 streaks, Cano has gone 27-79 since September 25, for a .342 batting average. The odds of Cano hitting .342 or better for 79 ABs are a mere 1 out of 3.6.
Finally, this perhaps does not need stating, but the emotional state of some observers makes me feel it necessary to clarify: the fact that Cano was hot in the end of the regular season and cold in the playoffs does not suggest Cano is a choker. Apart from these being small sample sizes, remember that those regular-season games were crucial - if Cano had played poorly, the Yankees might not have won the division and thus may never have advanced to the ALCS.
Technical note: I used .305 as the probability of Cano getting a hit in a given at-bat, because his career average is .305 including both the regular- and post-season.