Creating a 8x8 matrix with elements labeled by binary strings and entries following some probability distribution

I want to create an 8x8 matrix which provides the error probabilities in bit communication. The matrix looks as follows:

The columns amount to the observed quantities and the rows to the measured quantities. An element p[i,j] amounts to the conditional probability p(j|i). For example, the element p[0,1] gives the probability to observe the string 001 when the actual value is 000, i.e., it measures p(001|000).

Question: How can I create such a matrix in Python such that

1. The more bit flips there are the smaller is the equivalent conditional probability (for example p(100|000)<p(110|000)?
2. How to enable an "asymmetry". I.e., the probability of p(001|000)< p(000|001). That is, having bias that favors with higher probabilities transitions 1 to 0 than transitions 0 to 1.

Of course, the sum of probabilities in each row must equal to 1.

All in all, I want to create function in Python that takes as input an integer n (the size of the matrix, or equivalently where 2^n is the length of the bit string) and outputs a probability transition matrix with the above specified rules.

The difficulty is how to implement a probability distribution to fill the cells.

It is trivial to create an 8x8 array and fill diagonals:

P = np.zeros((8,8))
for i in range(8):
    for j in range(8):
        if i==j:
            P[i,j]=1

Similarly, its trivial to fill a given row or a given column by a fixed number. However, I cannot figure out (how to even begin) to fill such a matrix following the rules above, or even how exactly to define the distribution the elements must follow.

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