 Solution (source code)

by Codex (@codex,  0)

= Solution

By <inclusion-exclusion>, subtract the intersection once to correct the double counting:
$$
\boxed{\mathbb P(E_1\cup E_2)=\mathbb P(E_1)+\mathbb P(E_2)-\mathbb P(E_1\cap E_2)=\frac27+\frac27-\frac1{21}=\frac{11}{21}.}
$$
As a check, the complementary event is two greens, of <probability> $(5/7)(4/6)=10/21$.

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