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2 months ago Hájek, A. (2003). "What Conditional Probability Could Not Be." Synthese, 137(3), 273-323.
Alan Hájek critiques the quotient definition of conditional probability and explores alternative approaches, including the idea of using joint probabilities to provide a more intuitive understanding of conditional probability.
"The probability of two events transpiring concurrently (although not necessarily contemporaneously, as previously discussed in Example \ref{ex:concurrent_events}), given their respective conditional probabilities, is encapsulated by the formula $P \left( A \cap B \right) = P \left( A \mid B \right) P \left( B \right)$. This equation offers perhaps a more intuitive comprehension of the conditional probability concept. Indeed, there exist a number of authors who advocate for this interpretation to form the basis of the definition of conditional probability, as opposed to the quotient method."
Shall we provide a bibliographical reference to support the above claim? For example:
Fine, T. L. (1973). Theories of Probability: An Examination of Foundations. Academic Press.
In this book, Terrence L. Fine examines different foundational approaches to probability theory, including the possibility of defining conditional probability through the joint probability formula. He analyzes various interpretations of probability, some of which prefer using joint probabilities as a more intuitive basis for understanding the conditional relationship between events.