What is Conditional Event Algebra (requesting a simpler explanation than Wikipedia)?

Question

Is the idea behind Conditional Event Algebra to prove the formula for conditional probability, $\Pr(A\ |\ B) = \frac{\Pr(A\ \cap\ B)}{\Pr(B)}$, from the Kolmogorov axioms, departing from the standard approach of treating the conditional probability formula as a definition? I don't seem to have the prerequisite knowledge to follow the explanation on its Wikipedia page, but understanding the rationale behind conditional probability when not taken as a definition or an axiom is of interest to me.

If this is indeed what Conditional Event Algebra is for, I would like to ask for a high-level explanation of each type of Conditional Event Algebra listed on Wikipedia...

Shay algebras
Calabrese algebras
Goodman-Nguyen-van Fraassen algebras
Goodman-Nguyen-Walker algebras

I am currently following an introductory Probability Theory course, and have not studied Abstract Algebra in any detail, so I'm not sure if this material can be curated to the appropriate extent, but I would like to acquire some intuition about these approaches if feasible.

Answer 1

I just saw this and thought that as the person who created the Wikipedia article, I should take a shot at this answer. I’m returning to this topic after a long absence and so am rusty, but here goes.

I think the OP’s question is about motivation. Conditional event algebra (CEA) is not an attempt to justify the formula $\mathrm{Pr}(B | A) = \mathrm{Pr}(A \cap B) \div \mathrm{Pr}(A)$. Rather, that formula is assumed—but the standard use of it only gets you far. CEA tries to push further.

Let me try to explain using question-and-answer format.

Question: How can we talk about the probability that it’s snowing outside? Answer: We think of all the different outcomes (ways things might be outside), we assign each a probability, we add up the probabilities for all the outcomes that involve snow, and that’s your answer.

Question: How can we talk about the probability that it’s sunny AND snowing? Answer: Same deal, but now we also bring set intersection into the picture.

Question: How can we talk about the probability that IF it’s sunny, THEN it’s snowing? Answer: Hmm. Now we need conditional probability. We compute Pr(snow | sunny) using the above formula.

Question: How can we talk about the probability that if it’s sunny, then it’s snowing, AND if it’s snowing, then we’re going sledding? Answer: ???

CAE tries to answer that last question by creating formal representations of conditional events—events described in English by if-then sentences—for which negation (complement), conjunction (intersection), and disjunction (union) are meaningful operations.

Goodman-Nguyen-van Fraassen algebras are the most important type of CEA, which is why I did the Wikipedia article for them (and didn’t get to the others). With GNvF CEAs, the idea is that a conditional outcome is modeled as an infinite series of ordinary outcomes. The conditional event "If $A$ then $B$" is the set of all series where if you start at the front and read from left to right, you hit an $A$-and-$B$ outcome before you hit an $A$-and-not-$B$ outcome. Each series gets a probability. The probability of the event "If $A$ then $B$" is the sum of all the probabilities of "If $A$ then $B$" series. Finally, the probability of the event "If $A$ then $B$, and if $C$ then $D$" is the sum of all the probabilities of series that are both "If $A$ then $B$" series and "If $C$ then $D$" series.

Answer 2

The conditional probability $P_B(A) = P(A|B) $ is introduced as a "new" probability measure on the probability space $ ( \Omega, F, P )$ : it is verified that $ P(A|B) $ is actually a probability measure as it satisfies the three well-known properties: $P (A) $ is positive, $P ( \Omega ) = 1$ and $ \sigma$-additivity.

Conditional probability answers the question: if B has occurred, how likely is A?

Venn diagram

Let's imagine $ \Omega $ as a square that encloses two sets, A and B. To say that B has occurred means that the point falls in the region of B. Under this hypothesis it is natural to assume that the probability that A will happen, that is select a point inside the region A, is intimately linked with the extension of the intersection of region A with B: $ P( A \cap B ) $. So, $ Area (B) $ designates how easily B can be selected and we can define:

$ P(A|B) = [ Area (A \cap B ) / Area ( \Omega) ]/ [ Area (B) / Area ( \Omega) ] = Area (A \cap B ) / Area (B) $

once we keep in mind the cooncept between favorable outcomes and total outcomes.

K. Baclawski, M. Cerasoli, G. Rota Introduzione alla Probabilità Unione Matematica Italiana, Bologna 1984