Probability of neither A nor B

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You’re picking 2 balls. So the answer should be like Probability of picking the first ball times Probability of picking the second ball.

Your first answer follows this, but as you can see, the second answer only has one probability. That means it is a probability of some condition for picking only one ball. (Your second answer actually corresponds to the question “Picking one ball from 60 and the probability of that ball being either black or red”)