This question is inspired by Olbers' Paradox.
Imagine a universe $U$ shaped like $R^3$ with points $P$ randomly distributed in a Poisson fashion throughout, with density parameter $\lambda$ such that in any measurable set $S$ of volume $V$, the expected number of points in $S$ is $V$.
By "Poisson" I mean that (h/t @Rahul) for an observer who knows $\lambda$, given two disjoint measurable sets $S_1$ and $S_2$ in $U$, the distribution of the number of points in $S_1$ is independent of the distribution of the number of points in $S_2$.
My questions are:
- Can one even speak of such a distribution rigorously? (I believe so based on this discussion)
- Assuming such a distribution exists, if I choose an arbitrary line $L$ in $U$, must $L$ intersect at least one star?
- (If the answer to #2 is "no") Choose a random line $L$ than intersects the origin. What is the probability that $L$ intersects at least one star?
I believe that given an arbitrary ray (half a line), there is a probability of 1 that that ray will intersect a star, simply because the volume of the radius $r$ cylinder around the ray is infinite, so by the property of your distribution (which I'm also persuaded exists) you can say that there is an expected infinity of stars along it somewhere.
However, this only tells us that there are almost surely stars along any ray. It doesn't actually guarantee that there will be such a star. This is like saying "If you pick a point on a dartboard, the probability of a dart arriving at exactly that point is 0, but there's nothing stopping that outcome from occuring." For a broader discussion, read this http://en.wikipedia.org/wiki/Almost_surely