What is the probability of an event that is impossible? suppose that a probability is approximated to be zero based on empirical results. does this mean that the event is impossible?

Explanation:

For the first question, the mathematical (statistical, to be more precise) definition of a event that is impossible is that its probability is zero.

There are two extremes when you talk about probability: 0 and 1. 0 means that the event cannot occur in any way, while 1 is that the event is certain (it cannot not happen).

An example of an event with probability 0 is: men born in the Sun. This is an impossible event, and so there is no chance (probability 0) to find a man born in the Sun.

For the second question, when you deal with empirical results, this is with an experiment, the fact of having a probability zero as result of approximations does not make that an event be impossible; the event is unlikely (if the experiments were carried out properly) but yet it can happen.

For example, say you put 1,000 ballots in a bag with the number 3, together with 1 ballot with the number 9. Then, you perform the experiment of drawing a ballot 2,000 times. Suppose you draw the ballot with the number 9 two times. In that case, your approximated experimental probability of drawing a 9 will be 2 /2,000 = 0.001. Which you could round to 0.00. That may be valid but it cannot be taken as that the event of drawing the ballot with the number 9 is impossible, it is just unlikely.