Interpretation of frequentist probabilities Engineer here. This may be a dumb question, but lately I've been wondering about what to make of frequentist probabilities. By definition, the probability of an event is the number of times $k$ it occurs in an infinite number $n$ of tries $$p=\lim_{n\to \infty}\frac{k}{n}$$ Yet in practice, the number of repetitions will always be finite. Now let us assume we can choose between two bets ($1$ and $2$, with winning chances of $p_1=10\%$ and $p_2=50\%$ respectively (e. g. a roulette game with ten numbers where we can bet on a specific number or on even/odd). My first question is: If we play just once, does it actually matter which bet we choose? After all, if I go for bet $1$, I may be the one in ten persons who wins and if I go for bet $2$, I may be one of the five in ten persons who lose and there is no way of knowing what will happen... If we go one step further, one might argue that even if we play many (but obviously still a finite number of) times $n$, the probability of continuously losing $(1-p_{1/2})^n$ remains finite, so no matter which bet we choose, we may still end up as one of the "unlucky guys" (and again, we would have no chance of predicting if that will be the case)... That might lead one to conclude that frequentist probabilities are useless since any individual or group can always end up on the losing side...
 A: 
Now let us assume we can choose between two bets ($1$ and $2$, with
winning chances of $p_1=10\%$ and $p_2=50\%$ respectively (e. g. a
roulette game with ten numbers where we can bet on a specific number
or on even/odd). My first question is: If we play just once, does it
actually matter which bet we choose? After all, if I go for bet $1$, I
may be the one in ten persons who wins and if I go for bet $2$, I may
be one of the five in ten persons who lose and there is no way of
knowing what will happen...

Choice matters, definitely. More in general, among disjoint events, less probable event can happen. This is not strange, this is the world of probability. Therefore in gambling, you can win if you bet for less probable event and can lose if you bet for more probable. All gamblers know that. However if the potential gain is the same, all rational gamblers bet on the more probable event. It holds for any sensible definition of rationality.
The same story can be easily extended to multiple betting/events.
Moreover this story hold for any definition of probability. It has very few to share with weakness of frequentist approach.
