Answer
The probability is $\frac{7}{13}$.
Work Step by Step
We know that there are fifty-two outcomes in a $52$ card deck, so $\text{n}\left( \text{S} \right)=\text{ 52}$.
And there are four outcomes when the pulled card is a queen, so $\text{n}{{\left( \text{E} \right)}_{\text{a queen}}}=\text{ 4}$.
And the probability of getting a queen is given below,
$\text{P}{{\left( \text{E} \right)}_{\text{a queen}}}\text{ = }\frac{\text{n}{{\left( \text{E} \right)}_{\text{a queen}}}}{\text{n}\left( \text{S} \right)}\text{ = }\frac{4}{52}\text{ }$
And there are twenty-six outcomes when the pulled card is a red card, so $\text{n}{{\left( \text{E} \right)}_{\text{a red card}}}=\text{ 26}$.
And the probability of getting a red card is given below
$\text{P}{{\left( \text{E} \right)}_{\text{a red card}}}\text{ = }\frac{\text{n}{{\left( \text{E} \right)}_{\text{a red card}}}}{\text{n}\left( \text{S} \right)}\text{ = }\frac{26}{52}\text{ }$
There are two outcomes when the pulled card is a red queen card, so $\text{n}{{\left( \text{E} \right)}_{\text{a red queen card}}}=\text{ 2}$
And the probability of getting a red queen card is
$\text{P}{{\left( \text{E} \right)}_{\text{a red queen card}}}\text{ = }\frac{\text{n}{{\left( \text{E} \right)}_{\text{a red queen card}}}}{\text{n}\left( \text{S} \right)}\text{ = }\frac{2}{52}$
Thus, the probability of getting a queen or a red card is
$\begin{align}
& {{\text{P}}_{\text{a queen or a red card}}}\text{ = }{{\text{P}}_{\text{a queen}}}+\text{ }{{\text{P}}_{\text{a red card}}}-\text{ }{{\text{P}}_{\text{a red queen card}}} \\
& =\text{ }\frac{4}{52}\text{ }+\text{ }\frac{26}{52}\text{ }-\text{ }\frac{2}{52} \\
& =\text{ }\frac{4+26-2}{52} \\
& =\text{ }\frac{28}{52} \\
& =\text{ }\frac{7}{13}
\end{align}$
