I have a probability problem and would be glad if you can help me on this.
Description of the question:
There is a system claim.We generated 6 number pairs by measuring the full structure of this system from different aspects. The number pairs are: [ (a1, a2), (b1,b2), (c1,c2), (d1,d2), (e1,e2), (f1,f2)] Since they are derived from the same system, if anyone of these numbers changed, all the other numbers would have to change as well. To prove these numbers represent the measures of the claimed system but not random numbers (namely system claim is false), we test them with 4 independent functions (X, Y, Z, W). For a random number, the probability of success in any of these functions is 6%.
Question 1:
Let's say we tested all the 6 number pairs with all the 4 functions and observed 6 successes out of the total 48 trials. The initial question is if the probability of observing this is statistically significant (e.g. <0.05) or could this occur by chance? If the numbers were fully statistically independent and random, I can use Binomial test with p=0.06, 6 success, 48 trials and the resulting p-value is 0.066, which is slightly > 0.05 and I thus would conclude that those numbers are random.
However, this does not seem right and I might be missing one important point since if anyone of these numbers change all the 6 number pairs has to change too. Considering this point, what exactly is the probability of this observation (6 success, 48 trials with p=0.06)?
Question 2:
If there was an additional custom rule such that only one of the 4 functions can be successful on any of those 6 pairs, then what would be the probability of the same observation (6 success, 48 trials with p=0.06).
An answer with descriptions would be very helpful. Thank you.