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.
