Let A and B be two statements such that A is satisfied if, and only if, B is satisfied. Can we then say Pr{A} = Pr{B}?
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$\begingroup$ Use logic to answer it yourself are $B \to A$ and $B \land A$ the same statement? en.wikipedia.org/wiki/Boolean_algebra $\endgroup$– TimCommented Nov 27, 2023 at 14:17
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$\begingroup$ @Tim: What do you mean exactly with "are 𝐵→𝐴 and 𝐵∧𝐴 the same statement"? If A and B are equivalent, evidently 𝐵→𝐴 is true. I don't understand what you mean, sorry $\endgroup$– RobinCommented Nov 27, 2023 at 14:25
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$\begingroup$ The Axiom of Specification of set theory asserts these two statements determine the same event. Thus, you are asking whether the probability of an event equals itself. Equivalently, you are asking whether the probability of an event depends on how it is specified. I bet you can answer either one of these questions correctly! @Tim Your translation of this question into propositional logic is erroneous. $\endgroup$– whuber ♦Commented Nov 27, 2023 at 15:10
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$\begingroup$ @whuber, the first thing to say here is that A=B, the rest follows $\endgroup$– AksakalCommented Nov 27, 2023 at 15:27
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1$\begingroup$ @Aksakal However, we cannot conclude that the statements A and B are equal. It is only supposed they specify the same set. For instance, let A be the assertion "x is a negative real number and x is the square of a real number" and let B be the assertion "x is a real zero of the exponential function." Each can be shown to imply the other, despite such different formulations. $\endgroup$– whuber ♦Commented Nov 27, 2023 at 15:30
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Think of it in an intuitive way: if 2 situations only occur together; for example, if someone goes running exactly every Sunday, no matter the weather, ... , the person can say: A = 'I go running', B = 'It is Sunday'. The probability for A is now the same as B since going for a run implies it is Sunday and being Sunday implies going for a run.
So yes, if A <=> B, then P(A) = P(B).
