  We study the probability that certain laws are satisfied on infinite groups,
focusing on elements sampled by random walks. For several group laws, including
the metabelian one, we construct examples of infinite groups for which the law
holds with high probability, but the group does not satisfy the law virtually.
On the other hand, we show that if an infinite group satisfies the law $x^2=1$
with positive probability, then it is virtually abelian.
