Interpoint distances for two colours of points

Suppose red points form a one-dimensional Poisson spatial process with density λ_r. Suppose blue points are distributed similarly and independently with density λ_b. In the figure, λ_r=1 and λ_b=1.5.

What is the probability that two neighbouring points are the same or different colours?

Let r,b be the interpoint distance between red and blue points respectively. These RVs have pdf exp(-r/λ_r)/λ_r and exp(-b/λ_b)/λ_b. Let λ=λ_r+λ_b. The difference d=r-b thus has pdf (here [] is the indicator function)

(exp(-d/λ_r)[d≥0]+exp(d/λ_b)[d<0])/λ.

Integrating this, we get the simple final result: if I am a red point, my nearest neighbour on the right is blue if d>0, and Prob[d>0]=λ_r/λ.

Therefore

Prob[two neighbours are of different colours]

=Prob[red]Prob[neighbour blue]+Prob[blue]Prob[neighbour red]

=(λ_r/λ)λ_r/λ+ (λ_b/λ)λ_b/λ