The definition is not circular; sorry for the mistake. Another point bothers me: you did not agree with me when I said that a person must act in accordance with probability in the normative sense, just as he must not murder, and that the premise “a person should avoid dying,” for example, would not be enough to justify staying away from danger that might only possibly kill that person.
Suppose there were two lotteries, both with the same entry cost and the same prize amount, but the first has one hundred participants and the second has two hundred participants. A decides to enter the first lottery. Afterwards A is asked (by a questioner who assumes the premise “one should do what is required in order to make money”): “Why did you choose the first lottery and not the second? After all, just as it is possible that you will win the first, so it is possible that you will win the second.” A answers: “Because it is more probable that I will win the first lottery.”
According to the definition above, his words would have the following meaning: the claim “I will win the first lottery” is among the claims that will be true at a ratio of 1/100, whereas the claim “I will win the second lottery” is among the claims that will be true at a ratio of 1/200.
The problem is, seemingly, that the questioner does not assume the principle that one should act according to probability, and therefore that same questioner can ask again: why did you choose to act in accordance with the claim “I will win the first lottery,” which is true at a ratio of 1/100, rather than with the claim “I will win the second lottery,” which is true at a ratio of 1/200? After all, just as it is possible that the first claim is true, so it is possible that the second claim is true.

I would also like to understand more: in your view, when we claim that “it is more probable that someone will win a lottery with 10 participants than a lottery with 11 participants,” are we stating with certainty that if we repeat the two lotteries enough times, then a ticket from the first lottery will win more times, with certainty? Even if someone showed me that in fact this was not so after hundreds of thousands of repetitions or more, I would not retract that claim (“it is more probable that someone will win a lottery of 10 participants than of 11 participants”). Does this undermine the claim that there is a logical equivalence between these concepts (probability and a certain statement of the type mentioned above)?
And in general I would like to ask: is there a certain statement that is logically equivalent to the statement that a particular claim is probable?

As a continuation of the first comment: even if the Rabbi does not agree with me regarding “a judgment whose conditions are fulfilled as a logical basis for what one ought to do,” the Rabbi can still agree that a person can perceive a certain ideal that obligates him to act in accordance with probability; that is not what seemed to come out of the conversation we had.

The claim that a person must avoid dying is definitely enough to ground an obligation to stay away from danger. I never said otherwise.
As for the lotteries, the meaning here is much simpler: if you run many lotteries of the first type, you will win with a probability of one percent, and if you run many lotteries of the second type, you will win in half a percent of them.
There is no such thing as a person who does not act according to probability. Probability does not instruct a person how to act. Values instruct him. If he wants to make money, probability only shows him how to make money. If a person has a value not to use probability, that is a different discussion, and then indeed there is no criticism of his decision. I think this also answers your question in the next comment.
It seems to me that you are unnecessarily getting tangled up in simple definitions of probability.

I am not sure I understood the question about equivalence. The statement “claim X is true with a probability of 70%” is certainly true if claim X itself is true with a probability of 70%. Is that the equivalence you are looking for? It seems trivial to me.

I did not understand your remark in the last comment.

“Probability does not instruct a person how to act.” When I wrote “to act according to probability,” I meant that if there is a choice in which there is a one-in-three possibility that A will happen, and another choice in which there is a one-in-four possibility that A will happen, then it is incumbent on us, assuming we are interested in A, to choose the first option even if this is the last choice we will ever make (so that there is also no certain claim that we will achieve A more by this approach, at least not in the long run), despite the fact that there is no certain claim that we will achieve A. That is what I meant.
Even if we assume that a person is obligated to avoid death, in every situation in which there is a possibility that the person will not die, there will only be a “possible obligation,” unless we add to the normative claim the addition: from death or from what is likely to kill him. That is to say, only if the normative instruction itself refers to a case of mere probability, and not that it refers only to a certain case but that by virtue of the probabilistic situation (induction or probability) there is an obligation as if the person were definitely going to die. Even if we assume that whenever there is a probabilistic situation, then there is a claim about objective reality that if we repeat the event thousands of times then there will be a high number of occurrences of the event in question.

My question about logical equivalence is as follows: just as when I say there is a square in the room, I necessarily say that there is a shape in the room with four equal sides, in the same way, when I say that claim A is more probable than claim B, am I necessarily making a certain claim about reality (not that there is certainly a higher chance and not that there is certainly a universal rational tendency)?

The question that was bothering me was: what is the definition of probability? If we propose: a magnitude that is inversely related to the number of possibilities, or directly related to the number of times a phenomenon appeared under the same circumstances, then in the statement “A is probable” there is no certain conclusion external to the premise.
If we propose that the statement “A is more probable than B” entails the following statement: if we try both enough times, A will have an advantage over B, then that statement also has a certain meaning. But according to this proposal, it must be that something that would not show this phenomenon, even theoretically, would not be perceived as probable—and the question is, is that so? And in general one should ask: is it really true that if we try enough times, the above advantage will occur (for A over B)?

I did not understand your first argument. I defined the probability of a single case. The definition relies on what would happen in many cases, but it speaks about probability in a single trial.

When you say that the chance of getting an even result when rolling a fair die is one half, you have stated a certain claim (of course you cannot be sure that it is fair).

I did not understand.