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text: '$$\n\\newcommand{\\bP}{\\mathbb{P}}\n$$\n\n[summary: $$\n\\newcommand{\\bP}{\\mathbb{P}}\n$$\n\nKai 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: $\\bP(S \\mid \\neg D) = 0.2$. \n\nWe 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: \n\n$$\n\\begin{align}\n\\bP(L \\mid \\;\\;D,\\;\\;C) &= 0.4\\\\\n\\bP(L \\mid \\;\\;D,\\neg C) &= 0.1\\\\\n\\bP(L \\mid \\neg D,\\;\\;C) &= 0.7\\\\\n\\bP(L \\mid \\neg D,\\neg C) &= 0.9\n\\end{align}\n$$\n]\n\n\nWe 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 [-1rm] of $\\bP(D) = 0.4$ to our patient having the terminal illness.\n\nIt 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: $\\bP(S \\mid \\neg D) = 0.2$. \n\nIf 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 $(\\neg C)$. \n\nIn 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 $C$: \n\n$$\n\\begin{align}\n\\bP(L \\mid \\;\\;D,\\;\\;C) &= 0.4\\\\\n\\bP(L \\mid \\;\\;D,\\neg C) &= 0.1\\\\\n\\bP(L \\mid \\neg D,\\;\\;C) &= 0.7\\\\\n\\bP(L \\mid \\neg D,\\neg C) &= 0.9\n\\end{align}\n$$\n\nSo 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: $\\bP(L \\mid \\neg D,C) < \\bP(L \\mid \\neg D,\\neg C)$\n\nHere's a picture of the whole situation: \n\n<img src="http://i.imgur.com/TmFy2mB.png" width="648" height="523">',
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