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A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message is divided into blocks of a fixed number of digits. Then a single bit, called the "parity bit" is added to each block. The parity bit is $0$ if there is an even number of $1$'s in the block, and $1$ if there are an odd number of $1$'s. The received compares the block with the parity bit. If they do not agree, the block is retransmitted. If they do agree, the block is accepted as received.

The original block of, say, $n$ bits is transmitted as $n + 1$ bits. Suppose the blocks are three bits long—four, including the parity bit—and that the probability of mistransmitting a single bit is $0.1$.

Given that there is at least one error in the bit, what is the probability that it will be retransmitted?

Progress. I know that this scheme only detects an odd number of errors. There may also be an error in transmitting the parity bit.

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Hint: you will retransmit if there are $1$ or $3$ errors, not if there are $2$ or $4$ errors. Compute the chance of $1$ or $3$ errors and divide by the chance that there is at least one error.

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