Probability of Errors (parity)

2 dimensional even parity check matrix.
The rule is to make the total number of 1’s in the rows (and columns) even.

For example, in the first row, we have the message bits 1 0 0 1, there are two 1’s, so the parity bit is 0. The second column is 0 0 0 1, so the parity bit for that column is 1.

The question is:
If the probability of making a bit error (i.e. flipping a 1 to a 0) is p. What is the probability of undetectable errors?

For example, if there are two bit errors in the first row, the parity bit will still be 0, hence the errors are undetected.

1001 ]0
0001 ]1
0010 ]1
1110 ]1
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01001