Footnote: If you believe (as I do) that all probabilities about factual hypotheses are conditional probabilities [e.g., P(element 1| evidence x)], then the product rule, if applied at all, should be applied to such conditional probabilities (and not to freestanding, or unconditional, probabilities [e.g. P(element 1)].My own comment gave me the following thought(s):
Although the elements of claims or affirmative defenses -- more precisely stated, although the facts that satisfy the elements of legal claims or affirmative defenses -- may be "random variables" in the sense that they can take on various probability values, they are not random variables in the sense in which an evenly balanced die is a random variable.
The probability of an element (or, more precisely stated, the probability of a fact that satisfies an element) is conditional on evidence: i.e., P(Element n|Evidence x) -- or, more precisely stated, P(Fact n| Evidence x). That expression yields a probability: the resulting probability is a function of (i) the prior probability Fact n and, (on a Bayesian view of things), (ii) on the ratio of P(Evidence X|Fact n)/P(Evidence x|~Fact n). The same will be true for every other element of the claim or affirmative defense.
Is there any reason to assume that the chance of any probability value (between and inclusive of 0 to 1) for some Fact n given some Evidence x is equal to the chances of any other probability value between and inclusive of 0 and 1? I can't think of a valid reason why we should make this assumption. Can you? So a material fact in issue conditional on some evidence is not that sort of random variable: The correct analogue is not the equiprobability that we attach to each of the possible outcomes of an evenly-balanced die or an evenly-balanced roulette wheel, is it?
Note: Even the "prior probability" Fact n is not "absolute," or unconditional: The prior probability of Fact n is conditional on "background evidence," of evidence that is present, that is there, before Evidence x "comes along," or is considered. See Hajek's convincing argument that all probabilities are conditional probabilities.
If that's correct, just what, precisely, does the expression P(Fact n| Evidence x) represent or "stand for"? If the expression represents someone's epistemic uncertainty, ... whose uncertainty would a numerical value for P of Fact n given Evidence x represent? And once we have an answer to that question, what would our answer imply (if anything) for the chances of success for a proponent of a legal claim or an affirmative defense?
These are murky waters. Perhaps it is better for me to stay away from them. (I'm no mathematician or logician. I'm just a poor ol' befuddled law teacher.)