2c:null 2d:["$","title",null,{"children":"The Monty Hall Problem | Matt Diamond | Observable"}] 32:[] 30:{"id":"49f9ac453e0c4482","slug":"monty-hall-problem","title":"The Monty Hall Problem","thumbnail":"c1a15902fe5b789e8a0ca4553f40b82787589d2a992278963b43bee0b73040fa","access_level":"public","create_time":"$D2025-02-21T18:04:10.599Z","update_time":"$D2025-06-26T23:48:34.648Z","view_time":null,"first_public_time":"$D2025-06-26T23:26:17.027Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"356148d621554edd","login":"mattdiamond","name":"Matt Diamond","avatar_url":"https://avatars-next.observableusercontent.com/cb45f7f9a7385b5db6258b7130fd1d66bdb91af492cf965d26b049bda3d52391","tier":"starter_2024"},"version":345,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"f9e5784b8de6444e","login":"mattdiamond","name":"Matt Diamond","avatar_url":"https://avatars-next.observableusercontent.com/cb45f7f9a7385b5db6258b7130fd1d66bdb91af492cf965d26b049bda3d52391","tier":null}],"members":[{"type":"individual","id":"f9e5784b8de6444e","login":"mattdiamond","name":"Matt Diamond","avatar_url":"https://avatars-next.observableusercontent.com/cb45f7f9a7385b5db6258b7130fd1d66bdb91af492cf965d26b049bda3d52391","tier":null,"role":"editor"},{"type":"team","id":"356148d621554edd","login":"mattdiamond","name":"Matt Diamond","avatar_url":"https://avatars-next.observableusercontent.com/cb45f7f9a7385b5db6258b7130fd1d66bdb91af492cf965d26b049bda3d52391","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"The Monty Hall Problem","cells":[{"id":0,"value":"# The Monty Hall Problem\n## Why the nature of the host's choice matters\n\nThis notebook demonstrates that the correct choice in the Monty Hall problem is dependent on whether or not the host always chooses the door with the goat. If the host always selects a goat door, then it is beneficial to switch. However, if the host chooses randomly and just happens to have opened a goat door by chance, the benefit of switching disappears.\n\nThis is somewhat counterintuitive, as it appears that the best course of action depends on the host's knowledge and intentions, not the observable facts. However, it makes more sense when you consider that the probability of the host selecting the goat door by chance is higher if you've already selected the car. This impacts the probabilities in a way that erases the benefit of switching.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":142,"value":"viewof rerun = Inputs.button(\"Rerun simulation\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":111,"value":"viewof hostRandom = Inputs.radio([true, false], { label: 'Host door selection', value: true, format: x => x ? 'Random' : 'Goat' });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":262,"value":"\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

Strategy	Win Ratio
Always switch	${winRatioWhenHostPickedGoat(true)}
Never switch	${winRatioWhenHostPickedGoat(false)}

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"function pickDoors() {\n const doors = ['goat', 'car', 'goat'];\n const [myDoor] = doors.splice(randInt(3), 1);\n const hostIndex = hostRandom ? randInt(2) : doors.findIndex(d => d === 'goat');\n const [hostDoor] = doors.splice(hostIndex, 1);\n const [remainingDoor] = doors;\n return {\n hostDoor,\n myDoor,\n remainingDoor\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"results = rerun, repeat(pickDoors, 100000)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":86,"value":"function winRatioWhenHostPickedGoat(switchDoors) {\n const hostPickedGoat = results.filter(outcome => outcome.hostDoor === 'goat');\n const hostPickedGoatAndYouWon = hostPickedGoat.filter(outcome => {\n const finalSelection = switchDoors ? outcome.remainingDoor : outcome.myDoor;\n return finalSelection === 'car';\n });\n const winRatio = hostPickedGoatAndYouWon.length / hostPickedGoat.length;\n return winRatio;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"function randInt(n) {\n // returns random integer from 0 to (n - 1)\n return Math.floor(Math.random() * n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"function repeat(f, length) {\n return Array.from({ length }, f);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1"}} 31:"$30:body"