In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability larger than 5/8, must be abelian. In the talk I’ll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Be’eri Greenfeld, answering questions of Amir, Blachar, Gerasimova, and Kozma, and disproving a conjecture by Antolin, Martino, and Ventura.