What you end up with is not the real number system, however. Some of these work fine. As a result, the system you construct is not closed under addition, subtraction, multiplication, or division. I want to use 'divide by zero' to indicate a physically impossible task. Subtraction is the opposite of addition, as division is the opposite of multiplication.
Any number multiplied by zero gives zero.
In essence, the field axioms lay down a set of rules, i.e., basic assumptions, about how to put numbers together to get other numbers. As such, we disallow it as a valid operation on numbers. Division by 0 is not so much a phenomenon as it is a supposed construction, which is provably contradictory to a given set of rules and therefore not permitted within the system upon which the rules are imposed. To find the probability of something happening, you find out how many different ways the thing you're looking at can happen, and then divide that by the total number of things that can happen. In any case, the denominator can't be zero, since that means that there are zero things that can happen.