I'm afraid I don't understand your example at all. I would rather ignore the text of your post and try to answer the question in the title.
If you have a theory that makes statistical predictions and you wish to test it against data then it is broadly a question of whether the data matches what your theory predicts. There are different ways of measuring how good the match is.
If your theory predicts that all Fs are Gs, then a single counter observation of an F that is not a G is sufficient to falsify it. No amount of positive observations will prove it true, though if you repeatedly attempt to find counter observations and fail, you may say that your theory is corroborated.
If your theory predicts a particular proportion of Fs are Gs, the closer the observed proportion is to your prediction the better, though as before it is does not prove the theory is correct. It merely gives results that are expected in accordance with the law of large numbers.
There are other kinds of distributions which yield more complex comparisons. If a theory predicts that you will always observe one of two possible results then a Haldane distribution Beta(0,0) might be appropriate. Suppose a chemist synthesises a new pure compound and wishes to check whether it is soluble in water. One test is enough to tell you whether the compound is soluble or not. There is no evidential value in making repeated observations, unless perhaps you think you made a mistake.
In the case of prior probability distributions in Bayesian theory, a Beta(1,1) distribution is commonly used. In Bernoulli trials a Beta(0.5,0.5) might be more appropriate. In other applications, distributions such as Poisson, Erlang, or the old favourite Normal might be used.
Making observations does not always tend to increase the probability of future similar observations. Suppose you live next to a forest and every day you go foraging for mushrooms. You wish to form a statistical theory about how many mushrooms you will find and pick each day. It would be reasonable to suppose that there is a finite quantity of mushrooms in the forest and hence the more you pick the fewer remain. So the more observations (and pickings) you make, the less likely it is you will make future observations.
So there is no simple one-size-fits-all inductive procedure that tells you that some general proposition is more likely the more observations you make. You need to formulate specific models and test them. Ideally, testing them involves actively seeking out sources of sample bias, and comparing against rival models.