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15 March 2006
Probability and Causal Necessity
Given two events that have both occurred, for which probabilities can be assigned, where one of these events is necessary, and the other exists as a function of the necessary event, the event which is by far the most unlikely will tend to be the necessary event, and the other contingent. This is especially true if the gap between the two probabilities is statistically significant, as the probable can be molded to support the improbable far easier than the other way around.
Mathematical Formulation:
Given an extant event with probability n, and a second extant event dependant upon that first event with a probability kn, a ratio of stochastic balance exists of k:kn. Whichever side of this balance is smaller will tend to be the necessary event, and the other dependent, determine kn as the relative probability (assume n=1, k will be equal to the event's probability assuming n to be true.) Proof: both events exist; therefore can be assigned a probability. Given events a and b, where b requires a to exist in order to exist, hypothesis 1 would be that a is necessary, therefore given at a probability of 1. B would then be assigned a probability given a. Then b would be assumed to be true and a would be given a probability given b. As a's relative probability given b will always be 1:1, as b cannot exist without a, when we comparing these two relative probabilities, the relevant probability of b to a will be 1:b. If b is a stochastic event, the stochastic balance will be 1:(nb). If a's side of the equation is heavier, a exists as a function of b (even though b requires a,) and vice versa.
Lottery Example:
Positing a global lottery will allow us to test this theory. There is a global lottery not correlated with population, and there is a winner to that lottery. We can assign odds to this lottery, and clearly the lottery is logically (but not necessarily ordinally) necessary for the winner to exist. We can then use relevant probabilities to evaluate which of these two events are ordinally necessary, and which contingent. For our first lottery, we have odds of one in a million, and our n=6 billion. Looking at our stochastic balance, we see 1:1000, given the lottery, there should be about 1,000 winners, so the fact that someone won is not surprising. We can attribute the winner to random chance (if they didn’t win, someone else would have,) and we call the lottery necessary and the winner contingent. For our second lottery, let's posit odds of one in a trazillion. Our stochastic balance is 1:(1/zillion.) The fact that someone won is surprising, under random chance, no one should have won at all. Instead of the winner just being that lucky, the lottery was probably rigged for them to win. Therefore, the lottery was for the winner, rather than the winner for the lottery: the winner is necessary and the lottery consistent.
Anthropic Argument, Terrestrial Formulation
Looking at the existence of the Earth, we see a similar example to our lottery example. The universe's existence is our logically necessary event. The existence of a life-supporting planet is our second event. Our sample is all planets in this universe. Originally, we assumed that the probability of a life-bearing planet was relatively high, given the number of planets. The Anthropic Principle has shown that life-bearing planets are statistically very rare, on the order of 10^121. Generous estimates of the number of planets in the universe are in the range of 10^60. The stochastic balance then favors the extremely unlikely existence of a life-bearing planet as the necessary event, and the universe as the contingent event. This, of course, begs the question that if the lottery was fixed, then who fixed it and why? Which is, of course, a metaphysical question outside the bounds of this discussion, but regardless the answer is self-evident.
Statistics and Faith
As a sidebar, faith expressed in statistics is to take the object of faith as given, and to evaluate all other probabilities in reference to this given probability of one. Faith must assume a probability to be a certainty, but all probabilities are referenced to a certainty. Faith merely chooses which certainty should be used as a reference frame.
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