$$~$ \newcommand{\bP}{\mathbb{P}} $~$$

[summary: $$~$ \newcommand{\bP}{\mathbb{P}} $~$$

Kai has Dragon Pox with probability $~$\bP(D) = 0.4$~$. Patients with Dragon Pox sneeze sparks with probability $~$\bP(S \mid D) = 0.7$~$, while it is uncommon for healthy patients to sneeze sparks: $~$\bP(S \mid \neg D) = 0.2$~$.

We can treat Kai with the cure $~$(C)$~$ for Dragon Pox, or not $~$(\neg C)$~$. Then Kai lives $~$(L)$~$ or not $~$(\neg L)$~$, with probabilities depending on whether or not ve has Dragon Pox:

$$~$ \begin{align} \bP(L \mid \;\;D,\;\;C) &= 0.4\\ \bP(L \mid \;\;D,\neg C) &= 0.1\\ \bP(L \mid \neg D,\;\;C) &= 0.7\\ \bP(L \mid \neg D,\neg C) &= 0.9 \end{align} $~$$ ]

We are Healers in a hospital for magical maladies. Our patient, Kai, might be suffering from a bad case of Dragon Pox (denoted by the event $~$D$~$). Based on past experience, we assign a Prior probability of $~$\bP(D) = 0.4$~$ to our patient having the terminal illness.

It is well known that Dragon Pox causes sparks ($~$S$~$) to fly out of the patient's nostrils when they sneeze [Gunhilda of Gorsemoor, 1581]. Patients with Dragon Pox sneeze sparks with probability $~$\bP(S \mid D) = 0.7$~$, while it is uncommon for healthy patients to sneeze sparks: $~$\bP(S \mid \neg D) = 0.2$~$.

If we're lucky, sneezes will happen, and we'll be able to observe whether or not sparks come out of Kai's nose. Then we must decide whether to treat ver with the cure $~$(C)$~$ for Dragon Pox, or with nothing $~$(\neg C)$~$.

In lots of previous cases, we've made diagnoses and treatments, and then later found out whether the patient in fact had Dragon Pox, and whether or not the patient lives $~$(L)$~$. So we have some beliefs about what will happen to Kai, depending on $~$D$~$ and $~$C$~$:

$$~$ \begin{align} \bP(L \mid \;\;D,\;\;C) &= 0.4\\ \bP(L \mid \;\;D,\neg C) &= 0.1\\ \bP(L \mid \neg D,\;\;C) &= 0.7\\ \bP(L \mid \neg D,\neg C) &= 0.9 \end{align} $~$$

So for example, if Kai has Dragon Pox, then ve has a much better chance of living if we give ver the cure than if we do not: $~$\bP(L \mid D,C) > \bP(L \mid D,\neg C)$~$. On the other hand, if $~$\neg D$~$, then Kai has a better chance of living if we don't treat ver with the cure, which is dangerous by itself: $~$\bP(L \mid \neg D,C) < \bP(L \mid \neg D,\neg C)$~$

Here's a picture of the whole situation: