What is the probability a person sees a tree by looking out of the window? I'm familiar with the basic probability rules which problems come in the form of:

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*What is the probability of getting a head next by tossing a coin? 0.5

*What is the probability of getting a 3 if a number is chosen uniformly at random from the set {1, 2, 3, 4}? 0.25

How more complicated forms of questions are calculated? for example: what is the probability a person sees a tree by looking out of the window given there are 2 trees that could be seen from that window?
There are many things to consider:

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*The person's eye vision.

*Time of the day (and how many light bulbs are around in case it's after sunset)

*The direction at which he looks.

*How tall are the trees?

*Can the person pass a drug test at the given moment?

*Does he have a history of delusions?

And the list goes on which may contain infinite possibilities leading to seeing / not seeing a tree. Given we don't have a frequency of people who saw a tree by looking from that very same window, how to calculate the probability with any degrees of accuracy? How confident are we given that we are 100% confident the probability of getting 3 out of {1, 2, 3, 4} is 0.25? What if we don't have anything given in which case the question is the same as the title?

Well that's what statistics is about, right? All those variables you mentioned are unobserved and can impact the outcome, therefore we choose to encode this uncertainty about the problem as probabilities.
If you have no data, there is no way to answer the problem especially when probabilities are interpreted as relative frequencies. If you interpret probabilities as uncertainty about the problem as Bayesians do, then you can make a statistical conclusion based on that. For example in the absence of knowledge, you might assume that $Pr(tree) = 0.5$ which serves as your prior beliefs. Then whatever data you do observe, you update these beliefs and end up with some posterior beliefs. Obviously the more data you observe the better your estimates will be.
