What is the probability of a photon being emitted by the sun actually being absorbed by matter in the visible universe? In the Hubble ultra-deep field image it is apparent that the visible universe has a lot between us and our light horizon, but what is the likelihood that a photon emitted by our sun actually "hits" anything on its way to the bounds of our observable universe? 
 A: That likelyhood changes over time, in the early universe when everything was dense the mean free path of a photon was rather short, while in the meantime it has grown to approximately 10 billion lightyears, see

Paul Davies wrote: The mean free path is
about a Hubble radius (10²⁸ cm = 10²³ km).
Wikipedia wrote: The mean free path of a photon
is about 10²³ km, or 10 billion light years.

That distance will grow with time, since the density of the universe dilutes with its growing volume.
If your question is not in the cosmological but in the astronomical context, meaning that you are looking for the amount of the sunlight that hits the planets and their moons and the asteroid and Kuiper belt objects of the local solar system, you have to calculate their angular size in the frame of the sun and substract that from the 4π sterads of the celestial sphere, which I won't do here since there are a lot of small bodies to add up.
A: Since we really don't know how big the universe is what I suggest you do is take the size of the observable universe and then use the inverse square law along with the average photons the sun outputs in whatever given amount of time you want and then use the average density of matter in our universe and use all of those variables to calculate your answer.
