$\mathbf{\underline{Answer:\Rightarrow}}\;\;\mathbf{\underline{\bbox[orange, 5px, border:2px solid red]{\color{darkblack}{.5029}}}}$
$\mathbf{\underline{Explanation:\Rightarrow}}$
$\mathbf{\underline{Importance\; of \;word\; \color{blue}{"independently"} \;in\; the\; question:}}$
The word $\underline{\color{blue}{\mathbf{independent}}}$ here means that after selecting a number from the set of numbers, your count of number, that is, the sample space hasn't decreased.
In other words, it can be compared with the problem of picking a ball from the bag and then keeping it again in the bag. Then you can pick the next ball again from the same number of balls.
$\mathbf{\underline{Explanation:\Rightarrow}}$
Total numbers with $\mathbf{0}$ as the significant bits $=\mathbf 7$
Total numbers with $\mathbf{1}$ as the significant bits $=\mathbf 6$
Now,
The probability of picking the number with same $\mathbf{MSB0} =\mathbf{ \dfrac{7C_1\times7C_1}{13\times13}}$
The probability of picking the number with same $\mathbf{MSB1 = \dfrac{6C_1\times6C_1}{13\times13}}$
$\therefore$ Total probability $\mathrm{=\dfrac{7C_1\times7C_1}{13\times13}+\dfrac{6C_1\times6C_1}{13\times13} = \dfrac{49}{169}\times \dfrac{36}{169} = \mathbf{\underline{\bbox[orange, 5px, border:2px solid red]{\color{darkblack}{.5029}}}}}$
$\mathbf{\color{blue}{\underline{Binary\;representation\;of\;Numbers:}}}$
$\mathbf{NUMBER}$ 
$\mathbf{MSB}$ 


$\mathbf{LSB}$ 
0 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
0 
0 
0 
1 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
0 
0 
1

2 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
0 
1 
0 
3 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
0 
1 
1 
4 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
1 
0 
0 
5 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
1 
0 
1 
6 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
1 
1 
0 
7 
$\bbox[yellow, 5px, border:2px solid red]{\color{darkorange}0}$ 
1 
1 
1 
8 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
0 
0 
0 
9 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
0 
0 
1

10 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
0 
1 
0 
11 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
0 
1 
1 
12 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
1 
0 
0 
13 
$\bbox[blue, 5px, border:2px solid red]{\color{darkorange}1}$ 
1 
0 
1 
$\therefore$ The correct answer is $\mathbf{.5029}$