In short, "more likely" means a higher likelihood, which you can attempt to approximate using historical data, and which you can attempt to use in your favor when betting. Everything else below is offered to help provide a clearer way of thinking about these scenarios.
One source of confusion seems to be, "what if we don't know what the prior is" or "what if the prior is small". A can be more likely than B even if A and B are both very unlikely and even if I don't know how likely they are. For example, whats the chance that A: I win the lottery, vs B: the chance that I win the lottery AND am psychic? B is A with another condition so B < A no matter what. But both are unlikely, and I don't know what the numerical value of either is.
What does it mean for A to be more likely than B? For example, suppose
two people are throwing darts. The first person gets a bulls eye 6 out
of 10 times. The second person misses every single time by a wide
margin. Suppose now that A = first person is a professional dart
player and B = second person is a professional dart player.
Probability of First = Pro given that First scored 6/10 : P(Pro | 6/10)
Probability of Second = Pro given that Second scored 0/10. : P(Pro | 0/10)
You can calculate from Bayes law:
P(Pro | Score) = P(Score | Pro) P(Pro) / P(Score).
Both have the same prior P(Pro) so you really compare P(6 | Pro) / P(6) and P(0 | Pro) / P(0). If being a Pro means you will get 6s more often than an average person and 0s less often than an average person then:
P(6 | Pro) / P(6) = More > Less = P(0 | Pro) / P(0).
Is A more likely than B? If we look at priors, I suppose we can say
that a professional dart player is more likely to get 6 out of 10
bulls eyes than a non professional dart player. However, the opposite
implication doesn't necessarily hold. For example, a bad dart player
could get lucky, or a good but non professional dart player could also
get 6 out of 10 bulls eyes. Does this still mean that A is more likely
Yes, there may be unknown factors at play, but probability is a set of methods for dealing with our ignorance. So A is more likely than B on average, but some of the time even if you bet on the more likely outcome you'll be wrong. A coin might be weighted so that it lands heads 60% of the time. If forced to bet, you will bet heads. But 40% of the time, you will still be wrong.
If you imagine God's database of every time someone scored 6 and every time someone scored 0 and look at how often a 6-scorer was Pro and how often a 0-scorer was Pro, that takes into account all the possible scenarios and lucky days and unlucky days.
Or should this question be answered if we imagine a bet. For example,
if we knew that one of them is a professional dart player in advance,
which one would you bet on? When phrased in this way, it definitely
seems more rational to bet on A. However, we don't know in advance if
one of them is a professional dart player. As such, in what sense is A
more likely than B?
If you know one is a Pro, now you're asking a slightly different question.
P(A is THE pro | A got 6/10 and B got 0/10)
P(B is THE pro | A got 6/10 and B got 0/10) = 1 - above
Now it is like you look into a database of every time a Pro and Non-pro competed, and look at the times that the scores were 6-0. The chance of A being Pro is the fraction of the times that the player with 6 was pro in this database.
What about the scenario where there don't seem to be priors? For
example, suppose person A and B tries to guess where ten coin tosses
would land. Person A gets 3 coin tosses correct. Person B gets 9 coin
tosses correct. Is it more likely for person B to be a psychic than
person A? Well, if we go by priors, I'm not sure how there seems to be
any. In the case of the previous example, we have data that shows that
professional dart players atleast exist. In this case, we have no
prior evidence that suggests psychics exist.
Even without priors, since A and B have the same prior, when comparing which is more likely, you only compare the likelihood. Which is relative and based on how they scored.
Again, we can transform this into a betting scenario. If we knew in
advance that one of them was a psychic, what would you bet on? In this
case, again, it probably makes more sense to bet on person B. But not
only do we not know in advance that one of them is a psychic, we don't
know in advance that psychic abilities are even possible.
If you posit that you know one of them is a psychic that contradicts the part where you say we don't know if being psychic is possible.
So what does "more likely" mean in both of these cases? Or is this