how do we compute 4×4 matrix in binary symmetric channel

Question

I have question about binary symmetric channel (BSC), A sends i,j and B gets i,j as you can see in the picture:

If we want to show it using a 2×2, it will be like this (rows show A and columns B): $$ P =\pmatrix{ 1-p &p\\ p &1-p}$$

My problem is the way we should write this matrix for extension of degree 2. I have written this answer for it in my notebook, but I'm not sure about the solution.

$$ I = \pmatrix{ (1-p)^2 &(1-p)p &p(1-p) &p^2\\ (1-p)p &(1-p)^2 &p^2 &p(1-p)\\ p(1-p) &p^2 &(1-p)^2 &(1-p)p \\ p^2 &p(1-p) &(1-p)p &(1-p)^2} $$

My question is about the way we fill this 4×4 matrix. For example, how do we compute \begin{equation} p(B_{00}~|A_{11}) = p^2 \end{equation} ?

Answer 1

$p$ is the probability of error. Extension of two is to model transmission of two symbols. Hence there are four alternative cases.

Consider the first column (or row, since the matrix is symmetric): with probability $(1-p)^2$ both symbols are correctly received (no error). $(1-p)p$ and $p(1-p)$ correspond to error only on the first an second symbol, while the other is correctly received. $p^2$ is the probability of both in error.

You can see that each column (and row) sums up to one.

Answer 2

It's just the probability that both bits get flipped. Since they're independent it's just $p^2$