31:T8dc,md`## Answer Let + **E** be the event that the last passenger gets their seat + **q** the probability of **E** subject to a precondition There is a 1/100 probability that Prof. Tournesol (**T**) picks his seat, then the last passenger (and everybody else) gets their seat. Then **q**=1. There is a 99/100 probability that **T** picks another passenger's seat. This individual **X** boards the plane in 2nd, 3rd,... or 100th position (**p**), all equally probable possibilities, each with probability 1/99. + **p**=100: **X** must pick **T**'s' seat. Then **q**=0. + **p**=99: **X** has 2 choices. If he picks **T**'s seat (proba=1/2), then **X** and **T** have swapped seats and all others get their seats so **q**=1, else (proba=1/2) **q**=0 as this is like the **p**=99 case. Total: **q**=1/2. + **p**=98: **X** has 3 choices. If he picks **T**'s seat (proba=1/3), then **X** and **T** have swapped seats and all others get their seats so **q**=1, If he picks any other seat but that of the last passenger (proba=1/3) **q**=1/2 as this is like the **p**=98 case, else if he picks the last passenger's seat (proba=1/3) **q**=0. Total: **q**=(1/3)(1)+(1/3)(1/2)=1/2 + **p**=97: **X** has 4 choices. If he picks **T**'s seat (proba=1/4), then **X** and **T** have swapped seats and all others get their seats so **q**=1, If he picks any other seat but that of the last passenger (proba=2/4) **q**=1/2 as this is like the **p**=97 case, else if he picks the last passenger's (proba=1/4) seat **q**=0. Total: **q**=(1/4)(1)+(2/4)(1/2)=1/2 + More generally **p**=100-r with 31/2 `32:[{"id":"1e82b916033aae3fac4a11de162d978216d0a1f9d4999e851a5879ba8850ea62a5245a79965f354d532c23437f324e8ee5379c8b8eced0c48dc103b792ad0ac8","href":"https://static.observableusercontent.com/files/1e82b916033aae3fac4a11de162d978216d0a1f9d4999e851a5879ba8850ea62a5245a79965f354d532c23437f324e8ee5379c8b8eced0c48dc103b792ad0ac8","name":"Professor_Calculus.png","type":"image/png","lastModified":1613940887345.755,"size":31010}] 2f:{"title":"Professor Tournesol flies","cells":[{"id":0,"value":"md`# Professor Tournesol flies\n\nProf. Tournesol ([Calculus](https://en.wikipedia.org/wiki/Professor_Calculus) in English) boards a 100-seat plane with 99 colleagues. As he is the most famous of the cohort, he goes first. As he is a bit absent-minded he seats randomly. His colleagues follow him but are more disciplined so they seat where their ticket says except if the seat is taken, in which case they seat randomly.\n\n## Question\nWhat is the probability that the last one seats as planned ?\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"md`\n

${await FileAttachment('Professor_Calculus.png').image()}\n

Professor Tournesol aka. Calculus, by [Hergé](https://en.wikipedia.org/wiki/Herg%C3%A9)

\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":152,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":255,"value":"{\n const proba_f = new Fraction(proba).toFraction();\n\n return md`\n Total, proba(**E**) = (1/100)1+((99/100)(0+98(1/2)))/99 = **${proba_f}**\n `;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":192,"value":"proba = (1 / 100) * 1 + ((99 / 100) * (0 + 98 * (1 / 2))) / 99","pinned":false,"hidden":false,"mode":"ojs"},{"id":214,"value":"{\n const proba2_f = new Fraction(proba2).toFraction();\n\n return md`\n *NOTE*: In fact, the number of passengers does not change proba(**E**). \n\n For a number of passengers N=${N}, the formula becomes: \n proba(**E**) = (1/${N})(1)+((${N - 1}/${N})(0+${N - 2}(1/2)))/${N -\n 1} = **${proba2_f}**\n `;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":222,"value":"viewof N = Range([3, 200], { label: \"Nb passengers\", step: 1, value: 100 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":221,"value":"proba2 = (1 / N) * 1 + (((N - 1) / N) * (0 + (N - 2) * (1 / 2))) / (N - 1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"md`---\n## Annex\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":233,"value":"Fraction = require('fraction.js@4.0.13/fraction.js')","pinned":false,"hidden":false,"mode":"ojs"},{"id":225,"value":"import { Range } from \"@observablehq/inputs\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}