Derive the exact logical error probability of the three-qubit code

Derive the exact logical error probability of the three-qubit code

Under independent bit flips with probability p on each of three qubits, derive the logical error probability after majority decoding as a closed-form polynomial in p. Show every term. Deliverable: A derivation with the polynomial, the decoding rule, and a table of values for p from 0.0 to 1.0 in steps of 0.05.