= Solution
A <Poisson process> of rate $\lambda$ is a counting process starting at zero with independent stationary increments, with $N(t)-N(s)$ Poisson of mean $\lambda(t-s)$ for $t\ge s$, and right-continuous paths with unit jumps. Equivalently the small-interval probabilities are $\lambda h+o(h)$ for one event and $o(h)$ for two or more.
<Independence> of the two processes makes their joint increments over disjoint intervals independent. On an interval of length $h$, the sum of their independent Poisson increments has <probability generating function>
$$
e^{\alpha h(z-1)}e^{\beta h(z-1)}=e^{(\alpha+\beta)h(z-1)},
$$
so it is Poisson with mean $(\alpha+\beta)h$. Simultaneous arrivals have probability zero, so the sum retains unit jumps. It is therefore a <Poisson process> of rate $\alpha+\beta$.
The first arrival times are independent exponential variables of rates $\alpha$ and $\beta$. For positive rates,
$$
\boxed{\mathbb P(\text{first arrival red})=\int_0^\infty\alpha e^{-\alpha t}e^{-\beta t}\,dt=\frac\alpha{\alpha+\beta}}.
$$
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