The event 'A and B' is logically identical to 'B and A', so any valid decomposition of its probability must agree. This symmetry forces the equality of the two product formulas: P(A)P(B|A) = P(B)P(A|B).
Conditions: The argument uses commutativity of logical conjunction for events.; Both A and B have positive probability for the ordinary conditionals used here.
The event 'A and B' is logically identical to 'B and A', so any valid decomposition of its probability must agree. This symmetry forces the equality of the two product formulas: P(A)P(B|A) = P(B)P(A|B).
Conditions: The argument uses commutativity of logical conjunction for events.; Both A and B have positive probability for the ordinary conditionals used here.
In the square diagrams, the braces represent proportions of areas. P(A) is the fraction of the total sample space where event A occurs (a vertical strip).
Conditions: The visualization assumes probabilities can be represented by relative areas.; Both A and B have positive probability for the ordinary conditionals used here.
In the square diagrams, the braces represent proportions of areas. P(A) is the fraction of the total sample space where event A occurs (a vertical strip).
Conditions: The visualization assumes probabilities can be represented by relative areas.; Both A and B have positive probability for the ordinary conditionals used here.
Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.
Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.
Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.
Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.