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If A & B are joint, why can Arby recoup some of his loss only when $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$? But not $P_{Arby}(A \cup B) > P_{Arby}(A) + P_{Arby}(B)$?

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  1. Please see the sentence alongside the red line below. As the authors didn't write this sentence for the first case ( $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$), I'm assuming that if A & B ARE joint, Arby CAN recoup some of his loss. Correct?

  2. If I'm correct above, then why do these 2 cases differ? Scilicet, please see the question in this post's title. What's the intuition?

Image alt text Blitzstein, Introduction to Probability (2019 2 ed), p 40, Exercise 48. Selected Solutions PDF, p 7.

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