Given a $(n,M)$ code $\mathcal{C}_{n}$, its average probability of error is given by $$\begin{align*} P_e(\mathcal{C}_n):&=P(\hat W\not=W)\\ &=\sum\limits_{w=1}^{M}P(W=w)P(g(y^{n}\not=w|W=w))\\ &=\frac{1}{M}\sum\limits^{M}_{w=1}\lambda_{w}(\mathcal{C}_{n}) \end{align*}$$where $$\begin{align*} \lambda_{w}(\mathcal{C}_{n}):&=P(g(y^{n})\not=w|W=w))\\ &=P(g(y^{n})\not=w|X^{n}=f(w))\\ &=\sum\limits_{y^{n}\in\mathcal{Y}^{n}:g(y^{n}\not=w)}P_{Y^{n}|X^{n}}(y^{n}|x^{n}) \end{align*}$$is the code’s conditional probability of decoding error given that the message $w$ is sent over the channel.