Here's the principle: We compute one line after another, and to know the new living/dead state of a cell, we check the previous state of it and both its neighbours. Let's mark * a living cell and . a dead one. Then we have 8 possibilities (with a lot so symmetries though) according to neighbours:
     *** : center cell will live with probability p1
     *.* : center cell will live with probability p2
     **. : center cell will live with probability p3
     .** : center cell will live with probability p3
     ..* : center cell will live with probability 1-p3
     *.. : center cell will live with probability 1-p3
     .*. : center cell will live with probability 1-p2
     ... : center cell will live with probability 1-p1
The probabilities are linearily interpolated between the 3 values on top to the 3 values on the bottom (0 or 1 only on those extremities).