In recent years, probability theory has made a big impact on making progress on many long-standing problems in number theory, and many exciting new probabilistic problems have naturally emerged with strong arithmetic motivations. 

In this talk, I will tell you part of the story of their interactions. Many of the connections can be summarized as a random walk problem, but with influences from primes. 

You may find the story interesting if any of the following keywords catch your attention: the Ballot problem, Log-correlated fields, Gaussian multiplicative chaos, Fydorov-Hiary-Keating conjecture, Polya's conjecture, random polynomials, distribution of primes.