# Probability distribution of two one dimensional relativistic particles

I'm trying to solve the following problem (in preparation for my exam).
Consider two one-dimensional relativistic ideal-gas particles with masses confines to a one dimensional box of length $$L$$. Because they are relativistic, their energies are given by $$E_A=|P_A|c$$ and $$E_B=|P_B|c$$.

Assume that the particles are in thermal equilibrium with each other, and that the total kinetic energy is $$E=E_A+E_B$$. Use the usual assumption that the probability density is uniform in phase space, subject to the constraints.

Calculate the probability distribution $$P(E_A)$$ for the energy of one of the particles.

What I've done so far
This smells like a microcanonical ensemble. To find the probability distribution I use the equation

$$P(E_A)=\frac{\Omega(E_A)\Omega(E-E_A)}{\Omega{E}}$$

where

$$\Omega(E)=\frac{1}{h^{3N}N!}\int dq\int dp \delta(E-H).$$

We can write down $$H=c(|p_A|+|p_B|).$$

Then I try evaluating

$$\Omega(E_A)=\frac{1}{h}\int dq_A\int dp_A \delta\big(E_A-c(|p_A|+|p_B|)\big)$$

where $$h^{3N}$$ reduces down to just $$h$$ because we are looking at a single particle in one dimension. $$\int dq=L$$, and I reason that

$$\int dp_A \delta\big(E_A-c(|p_A|+|p_B|)\big)=2$$

because there are two values for which $$E_A-c(|p_A|+|p_B|)=0$$ since we take the absolute value of $$p_A$$. We can follow the same line of thinking for $$\Omega(E_B)$$ and afterwards substitute $$E_B=E-E_A$$. Then we find

$$\Omega(E_A)=\Omega(E_B)=\frac{2L}{h}.$$

This smells like total bullshit to me, since there is no dependency of $$E_A$$ anywhere and we are not involving the mass of particle A vs particle B. Nevertheless I can keep on trucking and calculate $$\Omega(E)$$ to be

$$\Omega(E)=\frac{\pi L^2E}{2h^2c}.$$

I'm pretty confident that this one is correct. However, this results in a probability distribution of

$$P(E_A)=\frac{2c}{\pi E}.$$

Surely, that can't be right!

My problem
Am I correctly evaluating the delta functions? It seems like there should be a dependency on $$E_A$$ or $$E_B$$, but this dependency is crushed by the one dimensional-ness of the problem. What is the correc way to reason through this?