Q:

What are the chances of a star colliding with another during a galactic collision?

My specific example for the question is the future collision of the galaxies Milky Way (our own galaxy) and Andromeda in a couple billion years. The star in question is obviously the sun in this case. I want to know the chances of a collision with another star and if it's significant or not. 

A:

Let's see what we get from some back-of-the-envelope estimates.
Imagine throwing one star (e.g., the Sun) at the other galaxy. How likely is it we'll hit a star in the other galaxy? Well, it's basically proportional to how big a target each star in the other galaxy is (its cross-sectional area) compared to the size of the whole galaxy, multiplied by the total number of stars in the target galaxy.
Let's assume it's the Milky Way-Andromeda scenario, so each galaxy has about 100 billion stars, and each star is roughly the same size as the Sun (some are much larger, most are smaller). The actual target area for an individual star is a circle with twice the star's radius (we're counting one star just grazing the other as a collision). Let's also assume the stars are more or less evenly distributed in a circular disk. Since "100,000 light years" is a common (and not completely crazy) estimate of the Milky Way's size, that's a circle of radius = 50,000 light years (about $10^{16}$ meters).
So: 100 billion stars in the target galaxy, each with target radius $\sim 2 R_{\odot}$, gives us a total target area of $10^{11} \times \pi (2 R_{\odot})^{2} \approx 10^{30}$ m$^{2}$.
The area of the target galaxy is $\pi R_{gal}^{2} \approx 10^{42}$ m$^{2}$. So the chance of our Sun hitting a star in the other galaxy is $\approx 10^{30} / 10^{42} = 10^{-12}$ -- or about one in a trillion.
The odds of any star from our galaxy not hitting a star in the other galaxy would be $(1 - 10^{-12})^{10^{11}} \approx 0.90$.
So there's only about a 10% chance of one (or more) of the galaxy's 100 billion
stars hitting a star in the other galaxy. And the chances of any one particular star (like our Sun) hitting a star in the other galaxy is about one in a trillion.

