the drunken man problem

The Original Drunken Man Problem

A drunk man is standing next to a pole at moment t_0=0.
At each moment t_s,k = k * tau the man takes a step in a random direction.
Find the most probable distance between the man and his initial starting position at time t.
For simplicity assume a constant tau = 1 second and all steps equal to 1 metre.

Elaborating on the Problem

An inebriated individual stumbles out a bar at moment t_0=0.
At each t_s,k = k * tau the guy takes a step in a direction given by angle alpha.
Every p steps alpha changes to a random direction then remains constant.
p follows a Poisson distribution with lambda = 4.
The distance travelled during one step follows a normal distribution with mu = 0.5 m and sigma^2 = 0.5 m^2.
To complicate matters further, upon colliding with the bar wall a new alpha is selected.
Find the most probable distance travelled at time t.
Then assume a normal distribution for tau too.