Let's consider the following events:
- $W$: the ground is wet,
- $R$: it rained this morning.
If, a priori, I attribute probability $0.5$ to event $R$, after learning that $W$ is realized, I will modify my beliefs according to Bayes' theorem.
$$P(R\mid W)=\frac 12\frac{P(W\mid R)}{P(W)}=\frac 1{2P(W)}$$
However, if I am not sure that $W$ is realized (let's say $P(W)=\frac 45$), for example if I have been told that the ground is wet without having seen it myself, how do I update my belief in $R$?
You would need to condition on the information that you do have; e.g.,
Then the updated probability is expressed as $P(R|J)$:
$$\begin{align}P(R|J) &=P(R|JW)P(W|J)+P(R|J\bar{W})P(\bar{W}|J)\tag{1}\\ &=P(R|W)P(W|J)+P(R|\bar{W})(1-P(W|J))\tag{2}\\ \end{align}$$ where, by Bayes' Theorem, $$\begin{align}P(R|W) &={P(W|R)P(R)\over P(W|R)P(R)+P(W|\bar R)P(\bar R)}={1\over 1 + {P(W|\bar R) \over P(W|R)}{1-P(R)\over P(R)}}\tag{3}\\ \\[2ex] P(R|\bar{W}) &=\cdots={1\over 1 + {1-P(W|\bar R) \over 1-P(W|R)}{1-P(R)\over P(R)}}\tag{4} \end{align}$$ assuming all denominators are nonzero.
Thus, your initial assessment of $P(R)$ is updated by Joe's statement $(J)$, using only your assessment of the three probabilities $P(W|\bar R), P(W|R),$ and $P(W|J)$.
Notes:
(1) is just the "law of total probability" for conditional probabilities. In the context of subjective probability, it is sometimes called "extending the conversation" to incorporate information not yet considered.
(1)$\implies$(2) assuming that if $W$ is given, then Joe's statement that $W$ is true is irrelevant to one's degree of belief in $R$; i.e., $P(R|JW)=P(R|W),$ etc.