This may be a silly question, but I cannot figure out a convincing (to myself) answer to it. Suppose that you want to buy a new car. Let $v$ be the value you attach to the car. Before visiting the dealer, you cannot tell for sure how much you value the car, i.e., you're uncertain about the true valuation of the car. However, you receive a observe the realization of a random variable $S\sim U[0,1]$, that tells you something about $v$: with probability $p$ the signal tells you the truth ($s=v$) and with probability $(1-p)$ the signal is noise, $s=\epsilon$, where $\epsilon$ is independent of $v$, and $\epsilon \sim U[0,1]$. This means that the expected value of $v$ given $s$ and $p$ is $\omega=ps+(1-p)\frac{1}{2}$.
Now, so long as $p>0$, the posterior of $\omega$ given $p$ and $s$ is: $ \mathbb{P}[\omega \leq x]=\mathbb{P}\left[s \leq \frac{x - (1-p)1/2}{p} \right]=\frac{x - (1-p)1/2}{p} $ because $s\sim U[0,1]$. My question is, how can I obtain the posterior when $p=0$? Any help to understand this is really appreciated it!