# Final Error of two module in serial and parallel

Suppose we have two modules in serial call them $M_1,M_2$ , a signal passing from $M_1$ to $M_2$ has probability of error $\epsilon_1$ , and at $M_2$ has $\epsilon_2$ probability of error.

In serial what is the final error probability of the signal?

What about in parallel?

What are the correct terminology to google this topic under?

PS : This is not a homework question as it would have been at least 20 years too late. :)

Thank you in advance

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## 1 Answer

For the series connection, the chance of no error is $(1-\epsilon_1)(1-\epsilon_2)=1-(\epsilon_1+\epsilon_2)+\epsilon_1\epsilon_2$so the chance of an error is $\epsilon_1+\epsilon_2-\epsilon_1\epsilon_2$. This is ignoring the possibility that two errors are made and cancel. If the error rates are small the error rate is about the sum of the two error rates.

For the parallel connection, what do you do with the two received messages? If you need either one to be correct (you have some checksum, for example, that identifies correct messages) then you only fail when both have errors, which happens $\epsilon_1\epsilon_2$ of the time. If the rates are small, this is smaller.

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