Isn’t every event by definition improbable in the sense that each
event precedes an infinite series of causes that could have
(theoretically atleast) been different?
Specific events don't have probabilities.
Probabilities express a relationship between measurements of initial conditions and expectations of future measurements. This relationship is based off of past measurements of one or more countable frequencies in one or more corresponding finite samples.
If you ran the experiment many times with the same measurements of initial conditions (and only the same measurements, not a metaphysically identical experiment in which all the things you don't know are also the same), what fraction of the outcomes will fall into the category being evaluated? That fraction is the probability.
We think of someone winning five lotteries as extremely improbable and
a rare event. But the probability of five different, specific people
winning five lotteries is the same.
Yes, if you remember to mentally translate the ambiguous tense of the present-continuous winning to a future tense. Common use of probability in the past tense to describe past events is misleading; the correct tense is the past progressive would have been.
When it comes to other types of mundane events, such as a person
walking across the street, isn’t that also an “absurdly improbable”
event? Once you factor in the details and specifics of that event,
such as the way in which the person is walking, what clothes he’s
wearing, what time it is, it also seems very improbable event.
So when, exactly, is an event probable?
When the "event" is defined as the category of outcomes of a hypothetical future experiment whose initial conditions have the values you (or your sources) have already measured, the event has a probability.
(Note all measurements have limited precision, so all predictable outcomes of measurements are categories of outcomes. If I predict a measurement of length on a meter stick with millimeter precision, I'm predicting a category of outcomes that would be measured within half a mm of the predicted value on a meter stick with greater precision.)
The probability of the category of outcomes is the ratio of how many hypothetical future experiments have outcomes that fall within the designated category, divided by the hypothetical large number of experiments whose initial conditions have the values you (or your sources) have previously measured. Making the designated category smaller or bigger makes the probability smaller or bigger respectively, which is what you seem to be getting at: for any experiment with a given set of initial condition measurements, the probability of a category of outcomes approaches zero as the size of the category becomes arbitrarily small.
If the "event" is not so defined, probability isn't one of the characteristics it can have.