The chance of Magikarp being Shiny is not yet known, but early research suggested 1 in 400. In the original games, Shiny Pokémon chance was determined by several factors:

From Generation III onwards, shiny Pokémon are determined by other factors such as the Trainer ID number and the personality value of the Pokémon. Despite this change, the odds of randomly encountering a shiny Pokémon is still 1 in 8192. – Source

There is no confirmation how rare a Shiny Pokémon is in Pokémon GO, but early research suggested that the chance of a Shiny Magikarp was 1 in 400 during the Water Festival event.

When Niantic releases more Shiny Pokémon, we’ll have more information and accurate values.