The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the desired properties with positive probability. For example: enter image description here https://cpb-us-east-1-juc1ugur1qwqqqo4.stackpathdns.com/sites.psu.edu/dist/f/7257/files/2013/10/The-Probabilistic-Method.pdf

Is there any example for geting some properties on planar graphs using probabilistic method. We know the discharging can get many useful local structure. I feel that the probability method is also a way of counting skill.