Yes, it is possible to have negative probabilities in some cases.
It is possible to have a negative probability?
First, for classical experiments, the probability for a given outcome on an experiment is always a number between 0 and 1, so it is defined as positive.
In some cases, we can have probability distributions with negative values, which are associated to unobservable events.
For example, negative probabilities are used in mathematical finance, where instead of probability they use "pseudo probability" or "risk neutral probability"
Concluding, yes, is possible to have a negative probability.
If you want to learn more about probability:
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