PokéBase - Pokémon Q&A
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If I changed an opponent's Pokémon's accuracy by n stages (e.g. using Flash |n|-times), where -6 <= n <= 0, and changed my Pokémon's evasivness by m stages (e.g. using Double Team |m|-times), where 0 <= m <= 6, then what is the chance, C, for the opponent's Pokémon's p accuracy move, where 50% <= p <= 100%, to hit for all combinations of n and m given a move with p accuracy?

In short, what is the chance I'll be able to dodge the opponent's Pokémon's move if I've lowered it's accuracy to the max AND raised my Pokémon's evasiveness to the max?

> "...a Pokémon with minimum accuracy attacking a target with maximum evasion will have no lower than a 33/100 or 33% chance to hit."

-Source: http://bulbapedia.bulbagarden.net/wiki/Stats#Core_series

Thank you Radicool21 and Dragonfree! :-)

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Radicool21's answer is only correct in the second-generation games. If there is both an accuracy modifier and an evasion modifier in the later games, the evasion modifier will be subtracted from the accuracy modifier, the result is capped at 6/-6 if it exceeds one of them, and the multiplier corresponding to that modified stat stage is used. In other words, the result is exactly the same whether the accuracy is -6 and the evasion is 6, the accuracy is -6 and the evasion is 0, or the accuracy is -3 and the evasion is 3.

So if the opponent's move's accuracy is 100, the modified accuracy will be simply 100 * 1/3 = 33. 1/3 is the multiplier corresponding to an accuracy/evasion stat stage of -6/6. You can find the rest in the Bulbapedia article Radicool21 linked to.

If you want it laid out in equations, first figure out the modified accuracy stat stage a:

a = min(6, max(-6, n - m))

Then the final accuracy of the move can be calculated as follows:

p * max(3, 3 + a)/max(3, 3 - a)
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The lowest your accuracy can go is 1/3 times normal, and the highest your evasion can go is 3x normal. (Source) Assuming the opponent's move has 100 accuracy, it is divided by 3, giving you 33 and 1/3. You must divide it by 3 again, giving you 11 and 1/9.

TL;DR about 11.1%