If one wishes to calculate the probability of some event (call it "X") which can happen in several distinct ways, one must add up for each separate case the probability of that case occurring, times the conditional probability that "X" occurs given that case.

For example, say we flip a coin twice and then roll one die for each "heads" that we got. So we have the following three cases:

1. Both flips are tails --> Don't roll any die. 
2. One tail and one head --> Roll one die. 
3. Both flips are heads --> Roll two dice.

Now given this scenario, suppose we want to calculate the probability of rolling a "6" on at least one of the die rolls. In words, the probability is given by the following:

The first factors in each term are given, respectively, by: zero, 1/6, and 11/36. Furthermore P(case1) = 1/4, P(case2) = 1/2, and P(case3) = 1/4.

Putting it all together, the result is: P(at least one 6) = 23/144.

The general formula for this kind of thing is the following:

P(X) = [Sum_i] P(X|i)*P(i)               (%)

where {i} is the set of cases by which the result X might come about, and P(X|i) is the conditional probability of X given i. In words, "the probability of X is equal to the probability of X *given* case 1, times the probability of case 1 -- plus the probability of X given case 2, times the probability of case 2 -- plus ..." and so on for all cases.

What does any of this have to do with anything?

In TEW's most recent attempt to explain the DDC results, I believe the concept of conditional probability (illustrated above) has been mis-used. In this new explanation, Dr. Little separates four distinct cases, which correspond to the various waves which may stimulate the emission of a particle pair. He then calculates the conditional probability of simultaneously detecting particles with (rotated) polarizers set at A' and B'. For example, in "case 1" (which corresponds to the particle pair being emitted under the influence of waves with polarizations A and B), the conditional probability of detecting the particles at A' and B' is found to be:

P(A',B'|case1) = -cos(A'-A) sin(A'-B) cos(B'-B) sin(B'-A)

This is equation (3) in Little's essay. (Note that I'm using A,B in place of A_1, A_2 to avoid putting subscripts everywhere.)

Similar expressions for the other three cases are found. These are then added together to find (purportedly) the total joint probability, which matches the quantum mechanical prediction.

However, by simply adding the probabilities for the 4 cases together, equation (%) above has been violated. The correct way to calculate the total joint probability is to apply equation (%), i.e., to weight each of the separate-case-probabilities by the probability of being in that case. In the present context, this means multiplying P(A',B'|case1) by the probability of particles being stimulated into waves with polarizations A,B. This probability, according to Little's recent paper (and as discussed in more detail in Little's original 1996 paper) is given by sin^2(A-B). The probability for what I have called case 4 (where the particles are stimulated by waves at A+pi/2 and B+pi/2) is the same: sin^2(A-B). The probabilities for cases 2 and 3 are both cos^2(A-B).

(Actually, you might notice that these add up to 2, not 1. That is a problem, but one which probably can be fixed by simply renormalizing all the probabilities by a factor of 1/2. In any case, this particular problem is not the one I want to focus on.)

Clearly if one recalculates the total joint probability P(A',B') using the correct formula -- that is, weighting each factor from Little's essay by either sin^2(A-B) or cos^2(A-B) as appropriate for each case -- one will no longer reproduce the correct quantum-mechanical joint probability.

Little appears to have recognized the possible dubiousness of his procedure for summing up cases to get the overall joint probability, as he spends about a page near the end of his essay addressing this very issue. Here Little argues explicitly that simply adding each conditional probability (without the correct weighting factors) is justified:

"the probabilities determined by the waves ... are proportional to the number of particles following those waves, with the same proportionality factor for all four [cases]."

So what is Little's argument for this conclusion? He begins by taking what I have called case 1 (i.e., particles emitted into A,B) and showing that the probabilities for all final states (i.e., combinations of A',B') in which the pair might be detected add up to the same sin^2(A-B) factor that gave the probability of being emitted into that state originally. In short, he shows that probability is conserved during the period after the particles are stimulated.

However, if used as an argument that the correct conditional probability weighting factors do not apply, this is simply a non-sequitur. That probability is conserved after the particles are emitted, does not mean that one can simply ignore the dependence of the total joint probability on the probabilities of the individual cases. In short, one must use the correct formula: equation (%).

(Actually, the argument doesn't even work to prove conservation of probability, since the terms involved can take on negative values. If the correct number of pairs stimulated into a given case is, say, 17, one does not "confirm" one's predictions by showing that one will detect 27 pairs in one final state and -10 in another. That isn't confirmation; it's nonsense.)

If the preceding discussion is at all unclear, one can see the basic point by simply looking at a concrete case. To take the clearest example, assume the initial (pre-particle-emission) polarizer settings are given by A = zero and B = pi/2. Now, the probability of a pair being stimulated into "case 1" (i.e., A,B) is sin^2(A-B) = 1 (up to perhaps an overall factor of 1/2, as discussed above) while the probabilities for pairs being stimulated into cases 2 and 3 (i.e., A,B+pi/2 and A+pi/2,B) are zero.

But if there are never any particles stimulated into cases 2 and 3, then clearly one should be able to simply omit the expressions for case 2 and case 3 when summing to get the total joint probability. Yet obviously, if one does omit these cases, one no longer gets the correct quantum mechanical result (since the conditional joint probability derived in the essay for cases 2 and 3 is not zero for A=0, B=pi/2).

This shows that by using a fallacious (equal) weighting of the four cases, there is actually a dependence on A' and B' (the final polarizer settings) entering the final joint probability formula which comes from an empty case -- that is, a case which contributes zero detected particles. In other words, Little has actually smuggled in the very dependence (on the distant polarizer settings) he was attempting to avoid.

The latest attempt by TEW to explain the DDC results in a fully local manner therefore fails. If one takes the TEW model seriously and calculates the total joint probability by applying the correct formula for conditional probabilities, the result will be found to be in accordance with the Bell inequality -- and therefore in contradiction to the Innsbruck experiment. Those of us who understood Bell's theorem from the beginning knew this had to be the case. Despite the mistaken claims of TEW's advocates to the contrary, the DDC results combined with Bell's theorem prove that no purely local description is possible. These experiments constitute a direct observation of a new type of (superluminal) causation.--Travis Norsen

Copyright 2001 Travis Norsen/ Objective Science. All rights reserved. Permission granted to link to this article only; but, permission is not granted to republish it. 

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