The probability that a depolarizing channel doesn't affect the information is usually assumed to be $1-3p$, while, for convenience, it is affected with same probability $p$ by any Pauli operator $X,Y,Z$.
Stepping to the error correction paradigms, it seems to be common to work only with operators $X$ and $Z$ as they can describe also a $Y$ operator (which, however results into a suppressed probability of $p^2$).
On contrary if one embeds in the framework also a probability $p$ for the $Y$ operator, this opens to the possibility, to "enhance" the noiseless probability to $$(1-p)^3 + p^3 = 1-3p + 3p^2.$$
I'm not sure if this is in contrast with the depolarizing channel, which states that the noiseless probability should be only $1-3p$. However, I want to argue that such an extra addendum $3p^2$ is not in contrast with the channel model, as this states that no noise occurs with probability $1-3p$, which doesn't necessarily mean that, if all the noise operators occurs simultaneously, they do not affect the logic of the system, even though a noise occurred.
In conclusion, I wonder whether such a reasoning is correct or I'm missing some fundamental concept regarding channels.