It is time for some maths. Ohh yeah.
So you have a shiny male Typhlosion. According to Bulbapedia shinyness is based on IVs
HP: 0 or 8
ATTACK: 2, 3, 6, 7, 10, 11, 14 or 15
SPECIAL : 10
Now, we come to breeding. The baby gets its IVs from the parent of the opposite gender, so you need to get a female Vulpix.
The Defense IV is passed, while the Special IV has a 50% chance of remaining the same. Attack and Speed are determined at random, at an approximate 1/15 chance. HP has to be 0 or 8
So 1/2 X 1/15 1/16 X 1/15 1/16 X 1/7.5 1/8 = 1/3375. 1/4096 1/512.
It's 1/512, that's the final answer, the final one. Whew.
EDIT: I done goofed up, cheers Unknown Letter Q. I also have messed up on the HP DV. Pardon the constant editing.
Hope I helped.