If A & B are joint, can Arby recoup some of his loss only when $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$?

1. Please see the sentence alongside the red line below, but the authors didn't write this sentence for the first case ( $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$). Thus if A & B _ARE _joint, can Arby recoup some of his loss in this first class?

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?

3. Why does Arby deliberately transact to his loss and detriment? Is Arby stupid? Oughtn't Arby transact the other way to earn money? Please see the sentences alongside my blue line below. Instead, why doesn't Arby SELL an A-certificate and a B-certificate, and buy an A ∪ B certificate?

4. Same question as 3. Instead, why doesn't Arby buy an A-certificate and a B-certificate, and sell an A ∪ B certificate?

5. In general, $P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$. Thus I see nothing wrong with $\color{limegreen}{P_{Arby} (A ∪ B) \neq P_{Arby}(A) + P_{Arby}(B)}$. Then how does "This problem illustrates the fact that the axioms of probability are not arbitrary, but rather are essential for coherent thought"?

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

posted almost 3 years ago

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2y ago

DNB‭

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