There are 32 possible values (0-31) for each of the 6 stats, which makes it 1 in 32^6 or 1 **in 1,073,741,824**! Source.

The shiny chance in X & Y is **1/4096**

Pokérus has a **3/65,536** or approximately a 1 in 21,845 chance of being generated on a Pokémon, either wild or bred. Source.

First, we find the chance of having Pokerus and 31 IVs on a Pokemon, which is

(1/1073741824 * 3/65536) = **3/70368744177664**

and multiply that by the shiny chance,

(3/70368744177664 * 1/4096) = **3/288230376151711744**

and gather five of these Pokemon together in a horde, and with generous rounding we get:

(3/288230376151711744)^5 = **243/19892929456391465686215289925873000000000000000000000000000000000000000000000000000000000000000000000000000000000000000**

which equates to roughly

**.000000000000000000000000000000000000000000000000000000000000000000000000000000000000012215395451569638936599271271031%**.

No, I am not kidding.

*Source: Calculations and above sources.*