There are no negative probabilities.  

There exists in the [phase space formulation][1] of QM “quasi-probability” distributions $W(x,p)$ that are *somewhere* negative, but the probabilities that occur as a result of integrating such distributions are always non-negative.

To be explicit, 
$$
\vert\psi(x)\vert^2=\int dp\, W(x,p)
$$
(for instance) is obtained by integrating all momenta of the joint quasi-distribution $W(x,p)$, $\vert \psi(x)\vert^2$ agrees with the probability density computed from the Born rule, and is everywhere non-negative so that the probability of finding the system in some space interval is *always* non-negative.

To illustrate this, look at the following two figures: 

[![enter image description here][2]][2]
[![enter image description here][3]][3]

They show the same quasi-distribution function, which contains regions of negativity near the center, as illustrated in the right figure: the near the central "anti"-peak the quasidistribution is clearly negative.  However, integrating this quasi distribution along $p$ at $x=-2$, shown on the left figure by a red line, yields

$$
\int dp W(x=-2,p)=\vert\psi(x=-2)\vert^2 > 0 
$$
so even if there are regions of negativity the probability of finding
the system near $x=-2$, which is $\vert\psi(x=-2)\vert^2\,dx$, is of course positive.


  [1]: https://en.m.wikipedia.org/wiki/Phase-space_formulation
  [2]: https://i.sstatic.net/olhA9m.png
  [3]: https://i.sstatic.net/xS86fm.png