This question is based on a discussion with a 10-year old. So if it is not clear how to interpret certain details, imagine how a 10-year old would interpret them.
This 10-year old does not know about relativistic issues, so assume that we are living in a Newtonian universe.
In this model, our universe is homogenous and isotropic, with properties such as we see around us. Specifically, the density and size distribution of stars is what the current models say they are.
This universe has the same size as our observable universe, around 45 billion light years.
If we froze time, and took a plane through this universe, would this plane go through a star?
I cannot figure out if the chance of this happening is close to zero or close to one. I know that distances between stars are very big, so the plane is much more likely to be outside a star than inside a star, so my intuition wants to say that the chance is very small. But on the other hand, this plane will be very big... So based on that, my intuition says that the chance is close to one. I expect the chance to be one of these extremes, I would be very surprised if the chance were close to 50%...
Clearly, my intuition fails here. And I don't know how to approach this problem better (generating entire universes of stars and calculating if a plane intersects one of the stars takes too much time...).
Rough estimates are perfectly acceptable, I only want to know if the chance is close to zero or close to one!
Edit: Reading the comments/answers, I noticed that my reference to the 10-year old did not have the intended effect.
Some of the answers/comments focussed on how an answer to the title question could be explained to a 10-year old. That was not my question, and I was a bit surprised to see several people interpreting it that way. My question is the one summarized in the title.
And some of the comments were about the definition of observable universe, and that it necessarily would slice through earth because earth is in the center of our observable universe. I added the reference of the 10-year old to avoid such loopholes...
Rob Jeffries' and Accumulation's interpretation of the question was exactly what I meant, so their answers satisfied me.