If you transmit a sequence of bits over some line, errors may creep in. For instance, if the Bit Error Rate of this line is 1%, on average, every 1 out of 100 bits will flip to the opposite value.

For a single transmitted bit, this relates to a Bernoulli trial with $p=0.99$ for being a 'success' and $q=1-p=0.01$ for a failure.

If this same bit is transmitted over two identical lines, there are two pathways of getting a correct result: two correct transmissions or two wrong transmissions (flipping the bit twice). The probability, therefore, is $p*p+q*q$.

Following this sequence for multiple transmissions, we get:

$$ \begin{array}{r¦c¦c} \textbf{ }&\textbf{correct}&\textbf{wrong}\\ \textbf{1}&p&q\\ \textbf{2}&p^{2}+q^{2}&2pq\\ \textbf{3}&p^{3}+3pq^{2}&3p^{2}q+q^{3}\\ \textbf{4}&p^{4}+6p^{2}q^{2}+q^{4}&4p^{3}q+4pq^{3} \end{array} $$

The parameters in front of each term seem to relate to Pascal's triangle. Furthermore, the 'correct' descriptions are 'symmetric' in q (even exponents), while the 'wrong' ones are 'anti-symmetric' (odd exponents). Can somebody explain the connection, and is there perhaps a general function which describes this series?

My real question is the following: what if the bits are transmitted across different lines $i$ each with a different BER $p_i$. The table then becomes:

$$ \begin{array}{r¦c¦c} \textbf{ }&\textbf{correct}&\textbf{wrong}\\ \textbf{1}&p_{1}&1-p_{1}\\ \textbf{2}&2p_{1}p_{2}-p_{1}-p_{2}+1&p_{1}-2p_{1}p_{2}+p_{2}\\ \textbf{3}&4p_{1}p_{2}p_{3}-2p_{1}p_{2}-2p_{2}p_{3}-2p_{3}p_{1}+p_{1}+p_{2}+p_{3}&etc. \end{array} $$

Is there a general function which describes this behaviour? Can someone explain anything more about this series? Does this have a name?


In the case of the repeated Bernoulli trials with identical probabilities, I believe the connection between this concept and Pascal's triangle can be visualised using the following graph.

Pascal's pathways

If we start at the top and define 'correct' to be a step to the right (bit does not flip) and 'wrong' to be a step to the left (the flipping of a bit), we can see there are 4 pathways to arrive at a 'wrong' result where the bit has flipped only once.

Similarly, there are 6 ways to arrive at a 'correct' bit which has flipped twice within 4 transmissions.

Therefore, the total probability of getting a correct answer is the sum of alternating Pascal's numbers, starting at the far right, moving to the left, each time skipping one number, and multiplying by the appropriate $p$'s or $q$'s.

This is the link with the 'symmetry' and 'anti-symmetry' mentioned in the original post.


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