This is a great question, first of all.
Let's take your example of seeing a person riding a bike on a street.
Without any additional information, you can give an educated guess that the probability would be less than seeing a car pass by. But by how much? Of course, this guess is based on prior information that you have that transportation by car is more common than transportation by bicycle. Is it possible to quantify the probability?
Now let's say you have access to the following information: a) the distribution of bike ownership in the city, b) the distribution of mean popularity of the 4 available transportation methods, c) distribution of means of popularity of ridership by season, month, and time of day, d) the distribution of age brackets, e) the popularity of bike ridership by age bracket etc.
The list can go on and on. Any additional factual piece information that you add, will nest the probability in a set of conditions assumed to be true. These will increase the certainty of one of the possible outcomes occurring: a) seeing a person riding a bike on the street b) not seeing a person riding a bike on the street. That is to say, if the probability was initially 50/50, now it asymmetrically in favour of one of the propositions. Regardless of which way it tips, Shannon entropy (the mathematical measure of uncertainty) has decreased, and the certainty of one of the outcomes occurring has increased.
Why two probabilities? Because your event has two possible outcomes. If there were more possible outcomes, the distribution of 100%, usually normalized to a value of [0,1] range with 0 meaning no probability and 1 meaning certainty, will be along k partitions.
The number of k partitions is partially defined by the event and partially dependent on what one is looking to predict. You could distribute the probability across any number of arbitrary events, depending on what you're trying to predict. So to answer the question you posed to the other poster, yes the individuation of events is in some sense subjective.
Philosophically, there is ongoing debate about how events are individuated. For example, some philosophers contend that event individuation depends on natural kinds. This means that conceptual joints must relate to ontological joints that are intrinsic to mind-independent phenomena. For example, if I want to know the probability of whales feeding their young, the concept of whale corresponds to a natural kind and the predication "feeding their young" is an empirical event. Music subgenres on the other hand are not natural kinds as they depend on convention. The other view contends that event individuation can be as fine-grained as our linguistic posits allow. This seems to go hand in hand with your suggestion that the level of description of an event is in some respects arbitrary. However, there are formal constraints on how events are to be individuated. Generally speaking you should partition classes into mutually exclusive and exhaustive sets.
Probability theory is agnostic about these philosophical views since it only defines the mathematical form of probability. We can use Bayes' Theorem to nestle the probability of an event given a set of dependent conditions. The probability of an event A can be computed by conditioning it to a set of mutually exhaustive and exclusive partitions/events B(i) where i = 1, 2, 3...n and summing those probabilities: P(A) = ∑P(A|B(i))P(B(i)). Conversely and by corollary we get: P(A|B) = P(A&B)/P(B). This is the formula for conditional probability which can be used to derive Bayes' Theorem -- P(A|B) = (P(B|A)P(A))/P(B) --, wherein the likelihoods -- P(B|A) -- can be any number of classes (events with discrete or continuous outcomes) that constrain A.
For artificially constructed scenarios like games e.g. chess, the sample space is discrete and can be determined via combinatorics, depending on whether we are counting combinations or permutations and whether we're sampling with or without replacement. What about sample spaces for continuous variables? A continuous sample space is a sample space where some interval contains an infinite number of possible outcomes. For example, the position or distance of a basketball on the court. Remember that discrete sample spaces can be finite but also infinite, specifically countably infinite, where 'countably' means 'can be put on a 1 to 1 correspondence with the natural numbers' as defined in Cantors' Theorem of Transfinite numbers.
Talk about sample spaces, presupposes classical probability, wherein Bayesian probability can also be assimilated. In other words, classical and Bayesian probabilities presuppose probability distributions of variables, which means we can work with some estimation of the sample space in advance. However, there's another conception of probability called Empirical probability that eschews sample spaces and simply takes the ratio of outcomes over trials e.g. 10 tails over 25 tosses. Empirical probability does away with the a priori assumption of the principle of indifference where in binary scenarios like coin tosses for example the probability prior to trials is distributed equally between the outcomes i.e. 50/50.