Interpretations of Probability
At “Casting Lots” I tried to use the relationship between the field of statistics and Divine Providence as a means of looking at the verse â€œHipil pur hu hagoral â€” he cast a pur, that is a goralâ€ (Esther 3:7). I asked:
Which raises the more general question as to whether the Believerâ€™s lexicon even has meaning for the word â€œrandomâ€. Is anything truly random? How far does Providence extend â€” Only to those who know Him (Rambam)? Only to those who merit it? Only to humans? Or, as became mainstream thought in the Orthodox community since the idea was first introduced by the Gra and the Baal Shem Tov, that every event in history is providential? And if we do take the last stance, what does â€œrandomâ€ mean? What does a statistician study?
The question doesn’t exist according to the first positions on hashgachah peratis (individualized Divine Providence, hereafter: HP) that I mentioned. However, if every event, or at least every event that impacts any person’s life is the product of HP, then what random events do statisticians ever experience? What is it probability measures, if in truth Hashem causes everything witnessed according to His Plan?
(Tangent: For what it’s worth, my personal believe is that contemporary science supports the universal views of HP taught by the Gra and the Besh”t. Chaos Theory is usually popularized through the example of the butterfly in Africa that does or does not flap its wings, and could be the reason why there are or aren’t tornadoes ch”v in Kansas. Systems are complex, with feedback loops that allow small, often immeasurable, changes in initial condition to have large impact on the outcome. Even if only a small subset of people earn HP on a subset of the events in their lives, every event contributes to the set of causes for what happens to them. I don’t think partial HP is consistent with such a model of the world.)
The comment chain onÂ “Casting Lots” opened up a discussion of whether various understandings of the relationship between nature and Divine Providence wouldn’t actually rest on the various interpretations of probability. So, I hit Wikipedia as a quick-and-easy source of a list of the more discussed interpretations. Quotes will be from there.
1- Classical Interpretation
The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.
â€” Pierre-Simon Laplace, A Philosophical Essay on Probabilities
So, the probability of flipping a “heads” is 50% because there is nothing causing a heads more than a tails. Since there are two options, the probability of each is 1/2.
But if everything is ultimately caused by Hashem and reflects His Plan, then there are no events describable by the classical interpretation of probability.
(And for these reasons we can also dismiss probabilistic logic and propensity theory, both of which extend the classical theory’s use of the notion of cause.)
Frequency probability is the interpretation of probability that defines an event’s probability as the limit of its relative frequency in a large number of trials.
In other words, the probability of flipping a “heads” is 50% because given enough coin tosses, we will eventually find that half of them are heads.
I think this interpretation is the implication of Râ€™ Dessler saying that teva (nature) is merely a pattern that Hashem hides His action behind. (Although later he says that teva is a pattern those of us who donâ€™t merit miracles impose on the world of our perceptions. And those who experience miracles due so because their view of reality imposes a loftier pattern, one where justice and mercy have more reality than gravity or electromagnetism.)
Teva as opposed to seeing hashgachah, therefore inheres in the Law of Large Numbers, the idea that if you flip a coin more and more times, you will get closer and closer to 50% heads, and if you roll a die enough times, 1/6 of the rolls will be a 6, etcâ€¦
Therefore, I think REED is saying that Hashem chooses which coin-toss will be heads â€” and that is HP. But in order to hide His â€œHandâ€ from our sight, Hashem does so in accordance with the Law of Large Numbers.
3- Bayesian Probability
The Bayesian theory of statistics relates probability to knowledge and ignorance. To give Jaynesâ€™ formal definition of Bayesian Probability, â€œA probability p is an abstract concept, a quantity that we assign theoretically, for the purpose of representing a state of knowledge, or that we calculate from previously assigned probabilities.â€
For example, if someone puts 5 white balls and 5 black balls in a jar, and each person takes a ball, but no one looks until the 10th person takes their ball, each person has a 50:50 of opening their hand and finding a white ball. What if each person peeks, but doesnâ€™t share that information? If Iâ€™m the 2nd person taking a ball, then the person who took the first ball and knows it was white, also knows my odds are only 4 in 9.
That first ball-taker is assessing the probability of my picking a white ball, GIVEN that one white one was already taken. No one else has that given, they are assessing the simple probability of my taking a white ball out of the original even mixture. They are coming up with a different number, but really they are answering a different question. By changing the knowns, the givens, we alter what it is we are assessing the probability of, and therefore will get a different probability.
Similarly, the odds of my taking out a white ball after I looked in my hand and saw the ball was white is 100%. But thatâ€™s not â€œthe odds of my picking up a white ballâ€, full-stop. Itâ€™s, â€œthe odds of my picking up a white ball GIVEN that I picked up a white ballâ€.
Hashem has all the givens, so the only probability â€œquestionsâ€ He faces only events with known answers of 100% or 0%.
Using Bayesian Probability, saying that there is a 50% chance of flipping a heads (reminder: assuming HP applies) boils down to saying that we really can’t fathom the mind of G-d well enough to make any prediction about which way He would want the coin to fall.