The probability of photon collision

Question

I was reading a textbook and I couldn't figure out something that seemed really obvious: Assume that the space is uniformly and randomly filled with stars, and the mean radial distance between the stars is $l$. Photons are emitted from a stars's surface in all the directions and can collide with other stars. Now it says that the probability that a photon collides (for the first time) with a star with distance $r$ from the star that it came from, is $$P(r)=l^{−1}e^{−r/l}$$ and did not explain why at all.

Answer 1

By the way strictly what you have is a probability density function and you should not read it as $\displaystyle P(X=r)=\frac{e^{-l/r}}{r}$

but instead

$\displaystyle f(r)=\frac{e^{-l/r}}{r}$

Here we are talking about a continuous random variable and so we have a probability density function. So instead we need to talk about the probability that the first collision is between $a$ and $b$.

And we write $P(a<X<b)=\int^{b}_{a} f(r) dr$ Notice that when $a=0$ and $b=\infty$, we get the probability being 1 which is what you would expect.

I don't know how much probability theory you have done but the thing here is called an exponential distribution. Exponential distribution describes the first occurence of "a process in which events occur continuously and independently at a constant average rate." In this case, the matter in the universe is scattered uniformly, the probability of collision is always constant.

(The usual example of an exponential distribution is a light bulb failing where the chance of failing is always at a constant rate.)

https://en.wikipedia.org/wiki/Exponential_distribution