A photon travels in space for 10 billion years. What are the odds it will arrive here without interacting with a atom on the way? Space isn't a perfect vacuum and I wonder how an image of a galaxy can travel billions of years without becoming diffused by photon collisions with space matter?
 A: The thing about scatter is this: if a photon was traveling towards earth, and got deflected by even 1/10000th of a degree because of an interaction with an atom, it would no longer reach earth - it would miss by light years. So the image won't get fuzzy.
Instead there is a phenomenon called "extinction" which describes the attenuation (due to scatter or absorption) of light as it travels. This can be quite strong when light travels through interstellar gas clouds - you can observe this in changes in the spectrum (different wave lengths interact with different probability: you lose the blue light first).
If you say density of outer space is 50 atoms per cubic meter (assumed to be hydrogen), then the mass of the "pillar of universe" that light traverses in 10 billion years is 9.4E25 meters times 50 atoms times atomic mass unit = $7.8 kg/m^2$
On the surface of the earth the density of the atmosphere  is roughly 1 kg/m^3 so that long journey really does nothing - it's the occasional high density regions (like interstellar gas clouds) that will be responsible for the majority of absorption.
A: The photon can travel a long way before scattering. The Rayleigh cross section of hydrogen interacting with a photon in the visible region is about $\sigma=1.9 \times 10^{-32} m^2$. The distance a photon  travels before hitting the hydrogen is then $l=\frac{1}{n_e \sigma}$ where $n_e$ is the number of molecules per cubic metre. If $n_e=10^6 m^{-3}$ then $l=5.2 \times 10^{25}$ or about 5 billion light years. There are of course variations in the density of space, dust clouds etc and the cross-sectional area is highly dependent on wavelength.
The likelihood a photon after traveling a distance $x$ arrives without been absorbed is $T=\exp (-x/l)$. If the photon has travelled 10 billion years, $x=9.46 \times 10^{25}m$ so $T=0.16$. So the answer is about $16 \%$ (with some large error bars!) 
