$\mathbf{\underline{Answer:\Rightarrow}}\;\;\mathbf{\underline{\bbox[orange, 5px, border:2px solid red]{\color{darkblack}{.5029}}}}$
$\mathbf{\underline{Explanation:\Rightarrow}}$
$\mathbf{\underline{Importance\; of \;word\; \color{blue}{"independently"} \;in\; the\; question:}}$
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 $\mathbf{MSB-0} =\mathbf{ \dfrac{7C_1\times7C_1}{13\times13}}$
The probability of picking the number with same $\mathbf{MSB-1 = \dfrac{6C_1\times6C_1}{13\times13}}$
$\therefore$ Total probability $\mathrm{=\dfrac{7C_1\times7C_1}{13\times13}+\dfrac{6C_1\times6C_1}{13\times13} =\mathbf {\dfrac{49+36}{169} }= \mathbf{\underline{\bbox[orange, 5px, border:2px solid red]{\color{darkblack}{.5029}}}}}$
$\mathbf{\color{blue}{\underline{Binary\;representation\;of\;Numbers:}}}$
$\mathbf{NUMBER}$ |
$\mathbf{MSB}$ |
|
|
$\mathbf{LSB}$ |
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 $\mathbf{.5029}$