When the result of an experiment is a probability, the experiment itself generally involves multiple samples, so your repeating the experiment is not the recreation of a one-time event. If I had a sample size of several hundred, I can confidently say something about the several-hundred-and-first. Even when there are only seven or eight readings taken, you are talking about a probability based on accumulated data, which is warranted.
The case of meteorology here, is a red herring. These predictions are not based on an experiment, they are based on a model. "A 25% chance of rain" means that of the alternative simulations that the model generates, on average 25% of the land in your service area gets (simulated) rain. The model itself is improved continually to better simulate the environment. But there is never a decisive experiment where the model is found to either succeed or fail. Experiments in such a modelling science are of the form 'This tweak improves alignment with the observations when the altered model is run over previous time periods, that one does not.'
So the question does not apply in principle. At the point of application, either running the model over a range of inputs is taking multiple samples of tomorrow, or there is no experiment involved at all.
For me, your other example also has little to do with probability per se, but with the fact money is a bad proxy for real value. My reasons for not taking part in the game with the balls is just that 1) the utility of money (or anything else) is not linear, and 2) safety has a value of its own. I prefer safety over windfall gains, primarily because my life is poorly engineered and the cost of $1000 debt is much higher for me in emotional terms than the added utility of $1000 extra income.
Can either of these examples be tweaked to really ask what you are after?
What you may be after is the infamous 'ceteris paribus' assumption that all science makes. We have to assume observed probabilities behave something like mathematical probabilities unless there is a cause for them to deviate.
This is always questionable in principle -- we do know that some things really are just more random than that. (The direction things radiate from an isolated nucleus, for instance, is just not going to converge.) But it is particularly unlikely in practice, because we never know what might constitute a cause. Over time the ability to repeat our experiments is meant to ferret out the unnoticed causes. But that means all predictions, always, are potentially missing required premises.
Accepting science as a model just involves some articles of faith.