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What are the chances of this happening? Also can this occur during online battling or does this occur only in-game? With Max Affection that is.

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It doesn't seem to affect online as can be seen here. As for the evasion boost, I would assume that it is like a +1 boost, which multiplies the accuracy of the opponent by 0.75, so a 100% accurate move would hit 75% of the time. However, this is speculation as it is difficult to test evasion/accuracy boosts, because results can differ from theoretical probability.

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Thanks for giving me an idea of it. :)