Asking about probabilities makes no sense unless there is a probability distribution.

Unfortunately there is no reasonable distribution on R^{+}.

But an obvious distribution exists on interval [0,x], where x is any positive real number.

One thing we can do in such case is to try to solve the problem for [0,x] and then see

if the solution has a limit when we go with x to infinity.

With this approach it's obvious that the probability we look for equals 0.5 for every x,

so obviously it's limit is also 0.5 when x goes to infinity.