= Solution
A <Poisson process> of rate $\lambda$ is a counting process starting at zero, with independent stationary increments, such that an increment over an interval of length $t$ has the <Poisson distribution> of parameter $\lambda t$. It has right-continuous paths with unit jumps almost surely.
Independence of the two processes makes increments of $A+B$ independent on disjoint intervals. The sum of independent <Poisson random variables> with parameters $\alpha t$ and $\beta t$ is <Poisson> with parameter $(\alpha+\beta)t$, as follows by convolution:
$$
\sum_{j=0}^n e^{-\alpha t}\frac{(\alpha t)^j}{j!}
e^{-\beta t}\frac{(\beta t)^{n-j}}{(n-j)!}
=e^{-(\alpha+\beta)t}\frac{((\alpha+\beta)t)^n}{n!}.
$$
Thus $A+B$ is a <Poisson process> of rate $\alpha+\beta$. Its first red and green waiting times are independent <exponential random variables> of rates $\alpha,\beta$, so
$$
P(T_R<T_G)=\int_0^\infty\alpha e^{-\alpha t}e^{-\beta t}\,dt
=\boxed{\frac\alpha{\alpha+\beta}}.
$$
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