Hi Taka, I have to tell you, I admire your question. Native speakers tend not to notice these little quirks of the language. Slim chance means very little chance, almost no chance.
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That movie influenced the audience badly.
I don't think that 'badly' necessarily means something negative. Maybe, depending on its context, it could be 'influence strongly', right?
So if the sentence was 'That movie had a bad influence on the audience,' c