What is the effect of redundancy coding on the bit error rate?
The error probability in the channel is \$P_c = 10 ^ {-3}\$. There will be a bit error after the decoder if a 3-bit codeword contains two or three errors.

The probability that the codeword has two errors after going through the channel is:

$$ P_2 = \binom{3}{2} (1-P_c) P_c^2. $$

The term \$\binom{3}{2} = 3\$ is equal to the number of ways the channel can introduce two errors. If "c" stands for "correct" and "e" for "error", the ways to make two errors out of three bits are: "cee", "ece", and "eec".

The term \$1-P_c\$ is the probability that one bit is correct; the term \$P_c^2\$ is the probability that two bits are incorrect.

The probability that the codeword has three errors after going through the channel is

$$ P_3 = P_c^3. $$

There is only one way to make three errors ("eee"), and no correct bits.

So the probability that the repetition decoder gives you an incorrect bit is

$$ P_b = P_2 + P_3 \approx 3 \times 10 ^ {-6}. $$