The original Shininess Determination prior to Gen VI was XOR result (16-bit) < 8 (= 2 ^ 3), therefore you have to match 16 - 3 = 13 bits.
In XY however, there are reports of Shiny Pkxs whose XOR result falls between 8 and 15. This leads to the speculation of the threshold being raised to 16 (= 2 ^ 4), which will yield a 16 - 4 = 12 bit match.
The chances are thus 2 ^ -13 = 1/8192 and 2 ^ -12 = 1/4096 .
By the way, can we confirm the threshold by intentionally set up a breeding of non-Shiny with XOR result 16?