I have seen videos and they both say different odds?
Some of the simplified odds are rounded to the nearest whole number.
Base odds: 1/4096 With Shiny Charm: 3/4096 (~1/1365)
Masuda Method: 6/4096 (~1/683) With Shiny Charm: 8/4096 (1/512)
Consecutive fishing (max chain): 41/4096 (~1/100) With Shiny Charm: 43/4096 (~1/95)
DexNav: Complex mechanics; see Bulbapedia's table
Shiny Pokemon (Bulbapedia)