The probability of a dice landing on 6 5 times is 1/6^5. The probability of finding a rock that is shaped exactly like me purely by nature is unknown.
Yet I know for a fact that the latter should be more improbable. And I would say everyone would agree. But if I don’t know the probability, how do I justify this?
There's a nice concept called credence.
https://en.wikipedia.org/wiki/Credence_(statistics)
Credence is the subjective degree of belief in a proposition. It's useful when we can't sensibly put an objective probability on an event, but we can make our best guess.
One way to look at it is to ask what gambling odds you would consider to be fair regarding a given proposition.
Credence follows the same mathematical rules as probability. For example if two propositions are independent then the credence of their conjunction is the product of their individual credences, and so forth.
But we need not make metaphysical assumptions about reality, as we do with probability. We all have subjective opinions and beliefs, and credence lets us assign numeric values to those without having to make assumptions about what's true or real. To assign a credence to a proposition, all we have to do is interrogate our own subjective degree of belief.
The Wiki page I linked has more info, as does SEP.
https://plato.stanford.edu/entries/belief/
In your rock example, you might set a credence of .000000001 or so of finding a rock that looks just like you. It's not an objective probability. It's just a subjective degree of belief, or the gambling odds that you would consider fair in order to take the bet.
Do we need to "justify" probabilities? Sometimes the answer is "no", and that is the case for any sensible probability assigned to finding a rock that is shaped exactly like "Loyal Hurt". Common sense tells us that it is extremely unlikely† so if we are comparing it with a more every day event, like rolling a 6 on a fair die five times in succession, for all practical purposes it doesn't really matter what value we assign to the probability, as long as it is very small, e.g. 1/1000000000000. What practical decision rests on whether it is 1/1000000000000 or 1/100000000000 or 1/100000000000000?
Or you could adopt an inequality. I gather there is about 10^80 particles in the observable universe, so it would be difficult to argue that the probability is less than 10^-80. Above we argued that it is less than 10^-12, so we could expand that to an interval- the probability is somewhere between 10^-12 and 10^-80 (I suspect tighter bounds are not that difficult to come up with). Just because we don't know the exact value of the probability, doesn't mean we don't know anything about it.
Or we could adopt a distribution over the probability - a Beta distribution is useful for that as it is defined over the interval [0,1], expressing the fact that you don't know the true value of the probability, but you know some values are more plausible than others. The shape of the distribution expresses that knowledge.
Probability gives us a way of making models that describe properties of systems that we find hard to predict by treating them as random chance. Generally we are talking about deterministic systems, such as rolling a die, we just don't have enough information to predict the result - there is no actual randomness there in reality - just in our probabilistic model. This means probabilities are an epistemological rather than physical in nature.
So to sum up - when we assign a probability, we are always talking about a model of reality, not reality itself. Probabilities often represent degrees of belief (or "states of knowledge"). If they need justifying, then the justification comes from your model - so explain your model. Sometimes they are just subjective beliefs, which may not have a real model (e.g. gut feelings), which can't really be "justified", so just admit it is a gut feeling (Bayes rule is still valid). Sometimes they don't need justifying. Statements of the bleedin' obvious are self- justifying.
† Unless Loyal Hurt is a favourite subject of a large number of sculptors
Yet I know for a fact that the latter should be more improbable. And I would say everyone would agree. But if I don’t know the probability
You have just identified one cognitive capacity of human beings, and more generally of all animals with a natural brain. Animals don't survive in the wild, or in human-built environment, by computing probabilities according to the Theory of Probabilities, yet they all need to be able to assess, not exactly, but sufficiently accurately, the probabilities of at least some near future events. Without this cognitive capacity, they wouldn't be able to decide what to do next, and they would just die within a short time (and they don't).
how do I justify this?
The probability that a rock be shaped exactly like you depends on the criterion to decide that it is, so I don't think that there is any definite answer to this, but it would in any case come down to the quantity of information involved given the criterion.
Experience can be used to justify this, in many different ways.
Perhaps you have experience of seeing rocks. Even if you haven't seen 65 rocks in your life (that's only 7,776, so it's very possible you have!), you have seen a decent proportion of that number, and none of them were probably even similarly-shaped to yourself. If 1 in 7,776 rocks looked exactly like you, many more should look similar to you, based on your experience that rocks don't have drastically different shapes to each other. So your experience of seeing enough rocks, none of which look like you, tell you that this would be more improbable.
Perhaps you have experience of reading newspaper articles (or watching YouTube videos, etc.) about improbable events which have occurred. You might have seen some examples like "this cloud looks exactly like the virgin Mary", or so on, with a photo showing a cloud which does look quite like the virgin Mary. If the similarity in that case is notable enough to report, then it can't be a very common occurrence. On the other hand, "this man rolled five sixes in a row" doesn't match with your experience of things which were improbable enough that their occurrence was noteworthy. So, your experience of reports of improbable occurrences tells you that this would be more improbable.
Perhaps you have experience of seeing people, all of whom look different enough to tell them apart. If you have seen 7,776 people who all looked different, and you know there are many more people in the world, your experience justifies a belief that there are many more than 7,776 people who look different. So even if every rock looked exactly like some person, there would still be only a one in several billion chance that a given rock looked exactly like you in particular.
Note that none of these arguments justifies any belief that the probability of such an event is some particular number. Rather, they justify your belief that this probability is less than some other probability. It's usually a lot easier to compare two things and determine that one is less probable than the other, than it is to mathematically determine the exact probability of each event. But a comparison ─ that one probability is less than another ─ is all you have asked how to justify in this question.
