In 1987 Richard Feynman wrote an article arguing in favor for negative probabilities. The question should not be: “do they exist?” but rather: “are they useful?”. People have for centuries denied the existence of negative numbers – but thinking about negative money as debts is undeniably useful. How would you do that with negative probabilities? The idea is to loosen the requirement that probabilities must be strictly positive in every calculation and just require the end result to be positive. If you are interested, check out this blog post that has a very nice overview.
Just as a little teaser, here is a quote from Feynman’s article:
A man starting a day with five apples who gives away ten and is given eight during the day has three left. I can calculate this in two steps: 5 -10 = -5 and -5 + 8 + 3. The final answer is satisfactorily positive and correct although in the intermediate steps of calculation negative numbers appear. In the real situation there must be special limitations of the time in which the various apples are received and given since he never really has a negative number, yet the use of negative numbers as an abstract calculation permits us freedom to do our mathematical calculations in any order simplifying the analysis enormously, and permitting us to disregard inessential details.