Example: Dragon Pox

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

[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: