We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by $μ_k$ the uniform distribution on positive (dual) braids of length $k$, we prove that the sequence $(μ_k)_k$ converges to a unique probability measure $μ_{\infty}$ on infinite positive (dual) braids. The key point is that the limiting measure $μ_{\infty}$ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a by-product, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids.