Q&A: Considerations in Choosing a Hypothesis as Correct
Considerations in Choosing a Hypothesis as Correct
Question
Hello Rabbi, may the fast be meaningful and easy.
I wanted to ask the Rabbi how one chooses a hypothesis as the most correct one.
For example, I see a machine that produces a red ball out of a million black balls.
I do not know whether the machine operates randomly or not. Is there a formula for calculating what conclusion I should draw regarding the nature of the machine?
(By the way, there is no more than one trial…)
Answer
This is too general a question. Basically, you are asking to learn probability and hypothesis testing in a nutshell. If the machine produced one red ball out of a million black ones in a single trial, I would bet that it is not “fair,” like a die that lands on 5 a hundred times in a row.
Hypothesis testing in a nutshell: the criterion is probabilistic. You calculate the probabilities for your two hypotheses and choose the more probable one.
Of course, you need to define the possible hypotheses before the discussion begins (and that is usually not done with probabilistic tools, but based on familiarity with the circumstances and common sense).
Discussion on Answer
Assuming that “designed” means a machine that specifically chooses a red ball with certainty (there are of course other possibilities), then you need to calculate a conditional probability.
A – designed machine.
B – not designed.
You gave the prior probabilities:
P(A), P(B)
The conditional probabilities for the event that a red ball comes out are:
P(X/A) = 1
P(X/B) = 1/1000000
Now you need to calculate, using Bayes’ theorem, the reverse conditional probabilities:
P(A/X), P(B/X)
And comparing them gives you the answer.
Here it is obvious even without calculation that it is designed with a very high probability.
I am not familiar with the formula, but I plugged it in according to Wikipedia, and this is what I did:
(0.8/1000000 + 0.2 ) / 0.2
The result is 0.999996, and for the second one I got 0.00000399.
Does that make sense?
Absolutely. That is exactly the result.
If so, then the side according to which a universe containing human beings and dualism exists requires an explanation.
The chance that such a universe would come about randomly (according to the Stanford Encyclopedia) is 10^-229
The chance that such a universe would come about in a designed way is 1
As long as you see the hypothesis that there is a God as more possible and plausible than 9.9*10^-228, you are supposed to conclude that there is a God with 99% probability.
x/(x+(1-x)/(1e+229))=0.99
X=9.9*10^-228
Am I right?
Leaving aside the numbers, which are nonsense of course, the argument is correct. This is called the physico-theological proof, and I discussed it at length in my notebooks here on the site.
How do you calculate the probability of each one?
Suppose the likelihood, as I estimate it, that a machine is random is 80%, and the likelihood that it is designed is 20%.
The chance of getting a black ball is 10^-6.
Is there some formula to calculate which one I should prefer?