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I love pure mathematics, so much. However, I recently did probability and statistics and applied mathematics courses, and I'm confused with statistics. There's no proofs and logic in applied math courses. I've always wondered why $P(A \cup B)=P(A)+P(B)- P(A \cap B)$.

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    $\begingroup$ For the last statement, you should look at the Kolmogorov probability axioms. It is a straight forward proof once they are assumed. (And you know basic set theory) $\endgroup$
    – Adam
    May 28 at 21:54
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Applied mathematics can mean both proof-based approach to an applied subject, or to pure numerical and heuristic approach to problems.

So: both pure and applied mathematics can be proof-based. However, applied issues tend to be messier, as real world is messy and applications tend to come from there. Not that pure mathematics does not also get messy sometimes...

Whether you consider proof-based or more heuristic stuff easier is for you to decide, but it is a matter of personality and prior learning. Neither is inherently easier or harder. This seems to be your issue.

For the stochastics issue, you might want to search "probability of union", for example https://duckduckgo.com/?q=probability+of+union . If the answers do not satisfy you, you might want to find some lecture notes or a book that covers the basics of stochastics. It is sure to be discussed.

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