The answer is generally "events with zero probability do not happen," so observing one must be a contradiction, but that's using layman's speak. If you want to be more precise, you have to be very cautious with your definitions. It turns out you can observe one without a contradiction, but it is a corner case in the definitions of random variables.
One pattern which comes forth is when we argue that if we draw from a continuous distribution such as U(0, 1), and get, as an example, 0.31415, we can show mathematically that the probability of getting exactly 0.31415 was 0, because we take an integral of the probability density function from 0.31415 to 0.31415 which is always zero unless you include strange functions like the dirac delta function in your distributions. However, that definition involved a bit of circular logic. We first made our observation, and then we calculated some probability afterwards.
Probably the most useful construction I can think of to answer the question would be a random variable X, where the probability density function is a piecewise function:
- 1 if X is -1
- 1 if X is [0, 1] (inclusive of both endpoints)
- 0 elsewhere
What I've created is basically a uniform random variable between 0 and 1, but I added that funny discontinuity at -1. Now we can talk about the probability of any draw from X being negative. This is the integral of the probability density function from -infinity to 0, which is clearly 0, by the rules of integration. However, from the definition, it is clear that there is a negative number which can indeed be drawn. Thus observing a negative number does not contradict the definition of X.
Now one could explore the likelyhood of this event occurring. We can use tests like z-tests to show that events like this are infinitely unlikely, but that they do not contradict the original definition of X.
Now that's the math. The follow on question would be whether any particular philosopher who talks about "zero probability" is choosing to use the formal definitions from mathematics, or if they are using the words to describe something that's just a hair different. In general, I find philosophers who talk about "impossible" events are not intending to use those specific formal wordings. However, if they elected to use the precise words "zero probability," those are odd enough word choices as to suggest that they intended to use the formal mathematical definitions.
In the meta question related to this answer, Philip Klöcking brings up a few good points. One is that we have to consider alternate number systems besides those of the real numbers. However, if we consider other systems, we have to be careful to use their definition of "zero" carefully. Given that probabilities are defined on a metric space, and in metric spaces, if the distance between X and Y is "zero," then they are the same point.
The other is perhaps the more interesting philosophical question which I elided over by questioning definitions before focusing on the math. The question uses the phrasing "if one observes an event with zero probability." Within the language of probability, that phrase is well defined. Outside of that narrow scope, that phrase is tricky. Questions like "what is an event" and "can someone observe anything" are famously frustrating questions to answer without going in loops. The idea of an event having "zero probability" also involves mapping the real world into the mathematics of probability. How was that mapping done? I simply assumed it had been done meaningfully, due to the phrasing of the question, but there may have been a contradiction in how that was done. In the VSauce video How to Count Past Infinity, he explores a similar question around the 13 minute mark regarding the many mathematical concepts of infinity that have been invented.
If I may leave with one of my favorite puzzles, a game:
In this game there are two people, labeled Dealer and Player. The Dealer writes two different numbers down on two slips of paper, and seals them in envelopes. It doesn't matter what the numbers are. They can be 0, 5, 4081922, -382.393193, pi, anything. They just have to be different numbers. The Dealer then hands the two envelopes to the Player in any order they please. The Player then selects one envelope to open. They then must decide whether the number in the other envelope is greater or less than the number in the envelope that was just opened.
Obviously it's easy to win 50% of the time. The challenge of the game is to come up with a strategy which wins more than 50% of the time. Can you think of a strategy?
The solution is:
Open an envelope randomly. Before opening the envelope, you pick a random number from a distribution whose domain covers all of the random numbers. Any distribution will work, but a Gaussian distribution is the most usual choice. When you open the envelope, you compare this random number against the one that is chosen. If your random number is greater than the one you exposed, you say that the unseen number is the greater one. If the random number is smaller, then you say the unseen number is the smaller one. If they're the same, then pick randomly.
How does this work? Well let's look at what can happen. If the number you pick is greater or less than both numbers, then you're basically picking randomly, so you have a 50% chance. If you nail the number you see exactly, you have a 50% chance. However, if you choose a value which is between both numbers, you have a 100% chance of getting it right.
Because there's always a number between any two given real numbers, there's always a random value you can pick that happens to be between both numbers. Thus you can always win more than 50% of the time.
