{
  "index": "1985-B-4",
  "type": "GEO",
  "tag": [
    "GEO",
    "COMB",
    "ANA"
  ],
  "difficulty": "",
  "question": "Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly\non the circumference $C$ and another point $q$ is chosen randomly from\nthe interior of $C$ (these points are chosen independently and\nuniformly over their domains). Let $R$ be the rectangle with sides\nparallel to the $x$ and $y$-axes with diagonal $pq$. What is the\nprobability that no point of $R$ lies outside of $C$?",
  "solution": "Solution. Let \\( p=(\\cos \\theta, \\sin \\theta) \\) and \\( q=(a, b) \\). The other two vertices of \\( R \\) are \\( (\\cos \\theta, b) \\) and \\( (a, \\sin \\theta) \\). If \\( |a| \\leq|\\cos \\theta| \\) and \\( |b| \\leq|\\sin \\theta| \\), then each vertex \\( (x, y) \\) of \\( R \\) satisfies \\( x^{2}+y^{2} \\leq \\cos ^{2} \\theta+\\sin ^{2} \\theta=1 \\), and no points of \\( R \\) can lie outside of \\( C \\). Conversely, if no points of \\( R \\) lies outside of \\( C \\), then applying this to the two vertices other than \\( p \\) and \\( q \\), we find\n\\[\n\\cos ^{2} \\theta+b^{2} \\leq 1, \\quad \\text { and } \\quad a^{2}+\\sin ^{2} \\theta \\leq 1,\n\\]\nor equivalently\n\\[\n|b| \\leq|\\sin \\theta|, \\quad \\text { and } \\quad|a| \\leq|\\cos \\theta| .\n\\]\n\nThese conditions imply that \\( (a, b) \\) lies inside or on \\( C \\), so for any given \\( \\theta \\), the probability that the random point \\( q=(a, b) \\) satisfies \\( (1) \\) is\n\\[\n\\frac{2|\\cos \\theta| \\cdot 2|\\sin \\theta|}{\\pi}=\\frac{2}{\\pi}|\\sin (2 \\theta)|,\n\\]\nand the overall probability is\n\\[\n\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\frac{2}{\\pi}|\\sin (2 \\theta)| d \\theta=\\frac{4}{\\pi^{2}} \\int_{0}^{\\pi / 2} \\sin (2 \\theta) d \\theta=\\frac{4}{\\pi^{2}}\n\\]",
  "vars": [
    "x",
    "y",
    "a",
    "b",
    "\\\\theta",
    "p",
    "q"
  ],
  "params": [
    "C",
    "R"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "horizcoor",
        "y": "vertcoor",
        "a": "abscissa",
        "b": "ordinate",
        "\\theta": "anglevar",
        "p": "boundarypoint",
        "q": "interiorpoint",
        "C": "unitcircle",
        "R": "rectregion"
      },
      "question": "Let $unitcircle$ be the unit circle $horizcoor^2+vertcoor^2=1$. A point $boundarypoint$ is chosen randomly on the circumference $unitcircle$ and another point $interiorpoint$ is chosen randomly from the interior of $unitcircle$ (these points are chosen independently and uniformly over their domains). Let $rectregion$ be the rectangle with sides parallel to the $horizcoor$ and $vertcoor$-axes with diagonal $boundarypoint interiorpoint$. What is the probability that no point of $rectregion$ lies outside of $unitcircle$?",
      "solution": "Solution. Let \\( boundarypoint=(\\cos anglevar, \\sin anglevar) \\) and \\( interiorpoint=(abscissa, ordinate) \\). The other two vertices of \\( rectregion \\) are \\( (\\cos anglevar, ordinate) \\) and \\( (abscissa, \\sin anglevar) \\). If \\( |abscissa| \\leq|\\cos anglevar| \\) and \\( |ordinate| \\leq|\\sin anglevar| \\), then each vertex \\( (horizcoor, vertcoor) \\) of \\( rectregion \\) satisfies \\( horizcoor^{2}+vertcoor^{2} \\leq \\cos ^{2} anglevar+\\sin ^{2} anglevar=1 \\), and no points of \\( rectregion \\) can lie outside of \\( unitcircle \\). Conversely, if no points of \\( rectregion \\) lies outside of \\( unitcircle \\), then applying this to the two vertices other than \\( boundarypoint \\) and \\( interiorpoint \\), we find\\[\\cos ^{2} anglevar+ordinate^{2} \\leq 1, \\quad \\text { and } \\quad abscissa^{2}+\\sin ^{2} anglevar \\leq 1,\\]or equivalently\\[|ordinate| \\leq|\\sin anglevar|, \\quad \\text { and } \\quad|abscissa| \\leq|\\cos anglevar| .\\]\n\nThese conditions imply that \\( (abscissa, ordinate) \\) lies inside or on \\( unitcircle \\), so for any given \\( anglevar \\), the probability that the random point \\( interiorpoint=(abscissa, ordinate) \\) satisfies \\( (1) \\) is\\[\\frac{2|\\cos anglevar| \\cdot 2|\\sin anglevar|}{\\pi}=\\frac{2}{\\pi}|\\sin (2 anglevar)|,\\]and the overall probability is\\[\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\frac{2}{\\pi}|\\sin (2 anglevar)| d anglevar=\\frac{4}{\\pi^{2}} \\int_{0}^{\\pi / 2} \\sin (2 anglevar) d anglevar=\\frac{4}{\\pi^{2}}\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "lighthouse",
        "y": "sailcloth",
        "a": "pineapple",
        "b": "carpenter",
        "\\theta": "anchorage",
        "p": "windstorm",
        "q": "butterfly",
        "C": "caterpillar",
        "R": "stonework"
      },
      "question": "Let $caterpillar$ be the unit circle $lighthouse^2+sailcloth^2=1$. A point $windstorm$ is chosen randomly\non the circumference $caterpillar$ and another point $butterfly$ is chosen randomly from\nthe interior of $caterpillar$ (these points are chosen independently and\nuniformly over their domains). Let $stonework$ be the rectangle with sides\nparallel to the $lighthouse$ and $sailcloth$-axes with diagonal $windstorm butterfly$. What is the\nprobability that no point of $stonework$ lies outside of $caterpillar$?",
      "solution": "Solution. Let \\( windstorm=(\\cos anchorage, \\sin anchorage) \\) and \\( butterfly=(pineapple, carpenter) \\). The other two vertices of \\( stonework \\) are \\( (\\cos anchorage, carpenter) \\) and \\( (pineapple, \\sin anchorage) \\). If \\( |pineapple| \\leq|\\cos anchorage| \\) and \\( |carpenter| \\leq|\\sin anchorage| \\), then each vertex \\( (lighthouse, sailcloth) \\) of \\( stonework \\) satisfies \\( lighthouse^{2}+sailcloth^{2} \\leq \\cos ^{2} anchorage+\\sin ^{2} anchorage=1 \\), and no points of \\( stonework \\) can lie outside of \\( caterpillar \\). Conversely, if no points of \\( stonework \\) lies outside of \\( caterpillar \\), then applying this to the two vertices other than \\( windstorm \\) and \\( butterfly \\), we find\n\\[\n\\cos ^{2} anchorage+carpenter^{2} \\leq 1, \\quad \\text { and } \\quad pineapple^{2}+\\sin ^{2} anchorage \\leq 1,\n\\]\nor equivalently\n\\[\n|carpenter| \\leq|\\sin anchorage|, \\quad \\text { and } \\quad|pineapple| \\leq|\\cos anchorage| .\n\\]\n\nThese conditions imply that \\( (pineapple, carpenter) \\) lies inside or on \\( caterpillar \\), so for any given \\( anchorage \\), the probability that the random point \\( butterfly=(pineapple, carpenter) \\) satisfies \\( (1) \\) is\n\\[\n\\frac{2|\\cos anchorage| \\cdot 2|\\sin anchorage|}{\\pi}=\\frac{2}{\\pi}|\\sin (2 anchorage)|,\n\\]\nand the overall probability is\n\\[\n\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\frac{2}{\\pi}|\\sin (2 anchorage)| d anchorage=\\frac{4}{\\pi^{2}} \\int_{0}^{\\pi / 2} \\sin (2 anchorage) d anchorage=\\frac{4}{\\pi^{2}}\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalcoord",
        "y": "horizontalcoord",
        "a": "noncoordinate",
        "b": "antislope",
        "\\theta": "straightvalue",
        "p": "nonpoint",
        "q": "exteriorpoint",
        "C": "massivesquare",
        "R": "ovalshape"
      },
      "question": "Let $massivesquare$ be the unit circle $verticalcoord^2+horizontalcoord^2=1$. A point $nonpoint$ is chosen randomly on the circumference $massivesquare$ and another point $exteriorpoint$ is chosen randomly from the interior of $massivesquare$ (these points are chosen independently and uniformly over their domains). Let $ovalshape$ be the rectangle with sides parallel to the $verticalcoord$ and $horizontalcoord$-axes with diagonal $nonpoint exteriorpoint$. What is the probability that no point of $ovalshape$ lies outside of $massivesquare$?",
      "solution": "Solution. Let \\( nonpoint=(\\cos straightvalue, \\sin straightvalue) \\) and \\( exteriorpoint=(noncoordinate, antislope) \\). The other two vertices of \\( ovalshape \\) are \\( (\\cos straightvalue, antislope) \\) and \\( (noncoordinate, \\sin straightvalue) \\). If \\( |noncoordinate| \\leq|\\cos straightvalue| \\) and \\( |antislope| \\leq|\\sin straightvalue| \\), then each vertex \\( (verticalcoord, horizontalcoord) \\) of \\( ovalshape \\) satisfies \\( verticalcoord^{2}+horizontalcoord^{2} \\leq \\cos ^{2} straightvalue+\\sin ^{2} straightvalue=1 \\), and no points of \\( ovalshape \\) can lie outside of \\( massivesquare \\). Conversely, if no points of \\( ovalshape \\) lies outside of \\( massivesquare \\), then applying this to the two vertices other than \\( nonpoint \\) and \\( exteriorpoint \\), we find\n\\[\n\\cos ^{2} straightvalue+antislope^{2} \\leq 1, \\quad \\text { and } \\quad noncoordinate^{2}+\\sin ^{2} straightvalue \\leq 1,\n\\]\nor equivalently\n\\[\n|antislope| \\leq|\\sin straightvalue|, \\quad \\text { and } \\quad|noncoordinate| \\leq|\\cos straightvalue| .\n\\]\n\nThese conditions imply that \\( (noncoordinate, antislope) \\) lies inside or on \\( massivesquare \\), so for any given \\( straightvalue \\), the probability that the random point \\( exteriorpoint=(noncoordinate, antislope) \\) satisfies \\( (1) \\) is\n\\[\n\\frac{2|\\cos straightvalue| \\cdot 2|\\sin straightvalue|}{\\pi}=\\frac{2}{\\pi}|\\sin (2 straightvalue)|,\n\\]\nand the overall probability is\n\\[\n\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\frac{2}{\\pi}|\\sin (2 straightvalue)| d straightvalue=\\frac{4}{\\pi^{2}} \\int_{0}^{\\pi / 2} \\sin (2 straightvalue) d straightvalue=\\frac{4}{\\pi^{2}}\n\\]"
    },
    "garbled_string": {
      "map": {
        "x": "lugwzpro",
        "y": "hjfsqnev",
        "a": "rqxumbae",
        "b": "snzlfkic",
        "\\theta": "gvrtmaji",
        "p": "kmculfow",
        "q": "nytahsej",
        "C": "vtxqspae",
        "R": "ygralmiv"
      },
      "question": "Let $vtxqspae$ be the unit circle $lugwzpro^2+hjfsqnev^2=1$. A point $kmculfow$ is chosen randomly\non the circumference $vtxqspae$ and another point $nytahsej$ is chosen randomly from\nthe interior of $vtxqspae$ (these points are chosen independently and\nuniformly over their domains). Let $ygralmiv$ be the rectangle with sides\nparallel to the $lugwzpro$ and $hjfsqnev$-axes with diagonal $kmculfownytahsej$. What is the\nprobability that no point of $ygralmiv$ lies outside of $vtxqspae$?",
      "solution": "Solution. Let \\( kmculfow=(\\cos gvrtmaji, \\sin gvrtmaji) \\) and \\( nytahsej=(rqxumbae, snzlfkic) \\). The other two vertices of \\( ygralmiv \\) are \\( (\\cos gvrtmaji, snzlfkic) \\) and \\( (rqxumbae, \\sin gvrtmaji) \\). If \\( |rqxumbae| \\leq|\\cos gvrtmaji| \\) and \\( |snzlfkic| \\leq|\\sin gvrtmaji| \\), then each vertex \\( (lugwzpro, hjfsqnev) \\) of \\( ygralmiv \\) satisfies \\( lugwzpro^{2}+hjfsqnev^{2} \\leq \\cos ^{2} gvrtmaji+\\sin ^{2} gvrtmaji=1 \\), and no points of \\( ygralmiv \\) can lie outside of \\( vtxqspae \\). Conversely, if no points of \\( ygralmiv \\) lies outside of \\( vtxqspae \\), then applying this to the two vertices other than \\( kmculfow \\) and \\( nytahsej \\), we find\n\\[\n\\cos ^{2} gvrtmaji+snzlfkic^{2} \\leq 1, \\quad \\text { and } \\quad rqxumbae^{2}+\\sin ^{2} gvrtmaji \\leq 1,\n\\]\nor equivalently\n\\[\n|snzlfkic| \\leq|\\sin gvrtmaji|, \\quad \\text { and } \\quad|rqxumbae| \\leq|\\cos gvrtmaji| .\n\\]\n\nThese conditions imply that \\( (rqxumbae, snzlfkic) \\) lies inside or on \\( vtxqspae \\), so for any given \\( gvrtmaji \\), the probability that the random point \\( nytahsej=(rqxumbae, snzlfkic) \\) satisfies \\( (1) \\) is\n\\[\n\\frac{2|\\cos gvrtmaji| \\cdot 2|\\sin gvrtmaji|}{\\pi}=\\frac{2}{\\pi}|\\sin (2 gvrtmaji)|,\n\\]\nand the overall probability is\n\\[\n\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\frac{2}{\\pi}|\\sin (2 gvrtmaji)| d gvrtmaji=\\frac{4}{\\pi^{2}} \\int_{0}^{\\pi / 2} \\sin (2 gvrtmaji) d gvrtmaji=\\frac{4}{\\pi^{2}}\n\\]"
    },
    "kernel_variant": {
      "question": "Let\n\\[\nE=\\Bigl\\{(x,y,z)\\in\\mathbb{R}^{3}\\;:\\;\\frac{x^{2}}{9}+\\frac{y^{2}}{4}+z^{2}\\le 1\\Bigr\\}\n\\]\nbe the solid ellipsoid with semi–axes $3$, $2$, $1$ aligned with the coordinate axes.  \n\nChoose angles\n\\[\n\\theta\\sim\\mathrm{Unif}[0,\\pi],\\qquad \n\\varphi\\sim\\mathrm{Unif}[0,2\\pi),\\qquad \n\\theta,\\varphi\\ \\text{independent},\n\\]\nand put the (surface) point\n\\[\np=(x_{0},y_{0},z_{0})\n      =(3\\sin\\theta\\cos\\varphi,\\;2\\sin\\theta\\sin\\varphi,\\;\\cos\\theta)\\in\\partial E .\n\\]\nIndependently choose\n\\[\nq=(a,b,c)\n\\]\nuniformly from the interior of $E$ (with respect to volume measure).\n\nLet $B$ be the axis–parallel rectangular box whose two opposite vertices are $p$ and $q$.\n\n(a)  Find the probability $P$ that the whole box $B$ is contained in $E$.\n\n(b)  Given that $B\\subset E$, compute \n\\[\n\\mathbb{E}\\!\\left[\\operatorname{Vol}(B)\\mid B\\subset E\\right].\n\\]\n\n(c)  Given that $B\\subset E$, compute \n\\[\n\\mathbb{E}\\!\\left[\\Sigma(B)\\mid B\\subset E\\right],\n\\]\nwhere $\\Sigma(B)$ denotes the total length of the $12$ edges of $B$.",
      "solution": "Step 1 –  A correct containment criterion  \nWrite $p=(x_{0},y_{0},z_{0})$ and $q=(a,b,c)$.  \nBecause $B$ is axis–parallel, every vertex $v=(x',y',z')$ of $B$ is obtained by independently choosing\n\\[\nx'\\in\\{x_{0},a\\},\\qquad y'\\in\\{y_{0},b\\},\\qquad z'\\in\\{z_{0},c\\}.\n\\]\n\n($\\Longrightarrow$)  Assume $B\\subset E$.  \nIf, say, $\\lvert a\\rvert>\\lvert x_{0}\\rvert$, then the vertex $v=(a,y_{0},z_{0})$ of $B$ satisfies\n\\[\n\\frac{a^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}>\\frac{x_{0}^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}=1,\n\\]\nbecause the left–hand side increases strictly with $\\lvert x\\rvert $.  Hence $v\\notin E$, contradicting $B\\subset E$.  \nTherefore $\\lvert a\\rvert\\le\\lvert x_{0}\\rvert$.  \nExactly the same argument with the other coordinates yields\n\\[\n\\lvert a\\rvert\\le\\lvert x_{0}\\rvert,\\qquad\n\\lvert b\\rvert\\le\\lvert y_{0}\\rvert,\\qquad\n\\lvert c\\rvert\\le\\lvert z_{0}\\rvert .\n\\tag{1}\n\\]\n\n($\\Longleftarrow$)  Conversely, suppose (1) holds.  \nFor any vertex $v=(x',y',z')$ of $B$ we then have\n\\[\n\\frac{x'^{2}}{9}\\le\\frac{x_{0}^{2}}{9},\\;\n\\frac{y'^{2}}{4}\\le\\frac{y_{0}^{2}}{4},\\;\nz'^{2}\\le z_{0}^{2},\n\\]\nso that\n\\[\n\\frac{x'^{2}}{9}+\\frac{y'^{2}}{4}+z'^{2}\n\\;\\le\\;\n\\frac{x_{0}^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}\n=\\;1.\n\\]\nThus every vertex of $B$ (and hence all of $B$) lies in $E$; i.e. $B\\subset E$.  \nWe have proved\n\\[\nB\\subset E\\quad\\Longleftrightarrow\\quad \\text{the three inequalities in (1)}.\n\\]\n\nStep 2 –  Probability of containment for fixed $p$  \nFor fixed $p$, the admissible points $q$ form the rectangular box\n\\[\nR(p)=\\{(a,b,c):\\lvert a\\rvert\\le\\lvert x_{0}\\rvert,\\;\n                       \\lvert b\\rvert\\le\\lvert y_{0}\\rvert,\\;\n                       \\lvert c\\rvert\\le\\lvert z_{0}\\rvert\\},\n\\]\nwhose volume is\n\\[\n\\operatorname{Vol}(R(p))=8\\lvert x_{0}y_{0}z_{0}\\rvert .\n\\]\nSince $R(p)\\subset E$, the conditional probability is\n\\[\n\\mathbb{P}(B\\subset E\\mid p)=\n\\frac{\\operatorname{Vol}(R(p))}{\\operatorname{Vol}(E)}\n    =\\frac{8\\lvert x_{0}y_{0}z_{0}\\rvert}{\\tfrac{4}{3}\\pi\\cdot3\\cdot2\\cdot1}\n    =\\frac{\\lvert x_{0}y_{0}z_{0}\\rvert}{\\pi}.\n\\tag{2}\n\\]\n\nStep 3 –  Expectation of $\\lvert x_{0}y_{0}z_{0}\\rvert$  \nBecause $\\theta,\\varphi$ are independent and uniform,\n\\[\nx_{0}=3\\sin\\theta\\cos\\varphi,\\quad\ny_{0}=2\\sin\\theta\\sin\\varphi,\\quad\nz_{0}=\\cos\\theta .\n\\]\nHence\n\\[\n\\lvert x_{0}y_{0}z_{0}\\rvert\n      =6\\lvert\\sin^{2}\\theta\\cos\\theta\\sin\\varphi\\cos\\varphi\\rvert .\n\\]\nWith joint density $(2\\pi^{2})^{-1}$,\n\\[\n\\begin{aligned}\n\\mathbb{E}\\bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\\bigr]\n&=\\frac{6}{2\\pi^{2}}\n  \\int_{0}^{2\\pi}\\lvert\\sin\\varphi\\cos\\varphi\\rvert\\,d\\varphi\n  \\int_{0}^{\\pi}\\lvert\\sin^{2}\\theta\\cos\\theta\\rvert\\,d\\theta  \\\\\n&=\\frac{6}{2\\pi^{2}}\\cdot2\\cdot\\frac{2}{3}\n  \\;=\\;\\frac{4}{\\pi^{2}} .\n\\end{aligned}\n\\]\n\nStep 4 –  Answer to part (a)  \nUsing (2),\n\\[\nP=\\mathbb{E}_{p}\\bigl[\\mathbb{P}(B\\subset E\\mid p)\\bigr]\n  =\\frac{1}{\\pi}\\,\\mathbb{E}\\bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\\bigr]\n  =\\frac{4}{\\pi^{3}} .\n\\]\n\nStep 5 –  Conditional expectation of $\\operatorname{Vol}(B)$  \nFor fixed $p$ and $B\\subset E$, $(a,b,c)$ is uniform on $R(p)$, so\n\\[\n\\mathbb{E}[\\,\\lvert x_{0}-a\\rvert\\mid p,B\\subset E]=\\lvert x_{0}\\rvert,\\quad\n\\mathbb{E}[\\,\\lvert y_{0}-b\\rvert\\mid p,B\\subset E]=\\lvert y_{0}\\rvert,\\quad\n\\mathbb{E}[\\,\\lvert z_{0}-c\\rvert\\mid p,B\\subset E]=\\lvert z_{0}\\rvert .\n\\]\nThus\n\\[\n\\mathbb{E}\\bigl[\\operatorname{Vol}(B)\\mid p,B\\subset E\\bigr]\n      =\\lvert x_{0}y_{0}z_{0}\\rvert .\n\\tag{3}\n\\]\n\nUnconditioning and using (2)–(3),\n\\[\n\\begin{aligned}\n\\mathbb{E}\\bigl[\\operatorname{Vol}(B)\\,1_{B\\subset E}\\bigr]\n      &=\\mathbb{E}_{p}\\!\\left[\n        \\mathbb{P}(B\\subset E\\mid p)\\,\n        \\lvert x_{0}y_{0}z_{0}\\rvert\n        \\right]\\\\\n      &=\\frac{1}{\\pi}\\,\n        \\mathbb{E}\\bigl[(x_{0}y_{0}z_{0})^{2}\\bigr].\n\\end{aligned}\n\\]\nCompute the second moment:\n\\[\n(x_{0}y_{0}z_{0})^{2}\n      =36\\sin^{4}\\theta\\cos^{2}\\theta\\cos^{2}\\varphi\\sin^{2}\\varphi .\n\\]\nUsing\n\\[\n\\int_{0}^{\\pi}\\sin^{4}\\theta\\cos^{2}\\theta\\,d\\theta=\\frac{\\pi}{16},\\qquad\n\\int_{0}^{2\\pi}\\cos^{2}\\varphi\\sin^{2}\\varphi\\,d\\varphi=\\frac{\\pi}{4},\n\\]\nwe find\n\\[\n\\mathbb{E}\\bigl[(x_{0}y_{0}z_{0})^{2}\\bigr]\n    =\\frac{1}{2\\pi^{2}}\\cdot36\\cdot\\frac{\\pi}{16}\\cdot\\frac{\\pi}{4}\n    =\\frac{9}{32}.\n\\]\nTherefore\n\\[\n\\mathbb{E}\\!\\left[\\operatorname{Vol}(B)\\mid B\\subset E\\right]\n   =\\frac{\\dfrac{9}{32\\pi}}{P}\n   =\\frac{9\\pi^{2}}{128}.\n\\]\n\nStep 6 –  Conditional expectation of the total edge length  \nFor fixed $p$,\n\\[\n\\Sigma(B)=4\\bigl(\\lvert x_{0}-a\\rvert+\\lvert y_{0}-b\\rvert+\\lvert z_{0}-c\\rvert\\bigr),\n\\]\nso\n\\[\n\\mathbb{E}\\bigl[\\Sigma(B)\\mid p,B\\subset E\\bigr]\n         =4\\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr).\n\\]\nHence\n\\[\n\\mathbb{E}\\bigl[\\Sigma(B)\\,1_{B\\subset E}\\bigr]\n   =\\frac{4}{\\pi}\\;\n     \\mathbb{E}\n        \\Bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\n              \\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr)\\Bigr].\n\\]\n\nBecause of symmetry,\n\\[\n\\mathbb{E}\\bigl[X^{2}YZ\\bigr]\n      =6/\\pi^{2},\\quad\n\\mathbb{E}\\bigl[XY^{2}Z\\bigr]\n      =4/\\pi^{2},\\quad\n\\mathbb{E}\\bigl[XYZ^{2}\\bigr]\n      =\\frac{3}{4\\pi},\n\\]\nwith $X=\\lvert x_{0}\\rvert$, $Y=\\lvert y_{0}\\rvert$, $Z=\\lvert z_{0}\\rvert$.  \nAdding these expectations gives\n\\[\n\\mathbb{E}\n   \\Bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\n         \\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr)\\Bigr]\n   =\\frac{10}{\\pi^{2}}+\\frac{3}{4\\pi}.\n\\]\nConsequently,\n\\[\n\\mathbb{E}\\!\\left[\\Sigma(B)\\mid B\\subset E\\right]\n  =\\frac{\\dfrac{4}{\\pi}\\Bigl(\\dfrac{10}{\\pi^{2}}+\\dfrac{3}{4\\pi}\\Bigr)}{P}\n  =10+\\frac{3\\pi}{4}.\n\\]\n\n\\bigskip\nFinal answers:\n\\[\n\\boxed{P=\\dfrac{4}{\\pi^{3}}},\\qquad\n\\boxed{\\mathbb{E}[\\operatorname{Vol}(B)\\mid B\\subset E]=\\dfrac{9\\pi^{2}}{128}},\\qquad\n\\boxed{\\mathbb{E}[\\Sigma(B)\\mid B\\subset E]=10+\\dfrac{3\\pi}{4}}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.687077",
        "was_fixed": false,
        "difficulty_analysis": "• Dimension raised from 2 D to 3 D: containment must be checked for a rectangular parallelepiped inside a solid ellipsoid.  \n• Three independent angular and Cartesian integrations replace a single one; Beta–functions, trigonometric identities and multi–variable expectations are required.  \n• Parts (b) and (c) involve conditional expectations that demand careful separation of the randomness of p and q, higher mixed moments (second and third), and deft use of symmetry.  \n• The final answers are non-elementary constants (e.g. 9π²/128), showing that mere pattern spotting from the planar case is impossible.  \n• The solution chain uses geometry, measure theory, probability, multivariate calculus and special-function identities—considerably deeper than in the original or the prior kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let\n\\[\nE=\\Bigl\\{(x,y,z)\\in\\mathbb{R}^{3}\\;:\\;\\frac{x^{2}}{9}+\\frac{y^{2}}{4}+z^{2}\\le 1\\Bigr\\}\n\\]\nbe the solid ellipsoid with semi–axes $3$, $2$, $1$ aligned with the coordinate axes.  \n\nChoose angles\n\\[\n\\theta\\sim\\mathrm{Unif}[0,\\pi],\\qquad \n\\varphi\\sim\\mathrm{Unif}[0,2\\pi),\\qquad \n\\theta,\\varphi\\ \\text{independent},\n\\]\nand put the (surface) point\n\\[\np=(x_{0},y_{0},z_{0})\n      =(3\\sin\\theta\\cos\\varphi,\\;2\\sin\\theta\\sin\\varphi,\\;\\cos\\theta)\\in\\partial E .\n\\]\nIndependently choose\n\\[\nq=(a,b,c)\n\\]\nuniformly from the interior of $E$ (with respect to volume measure).\n\nLet $B$ be the axis–parallel rectangular box whose two opposite vertices are $p$ and $q$.\n\n(a)  Find the probability $P$ that the whole box $B$ is contained in $E$.\n\n(b)  Given that $B\\subset E$, compute \n\\[\n\\mathbb{E}\\!\\left[\\operatorname{Vol}(B)\\mid B\\subset E\\right].\n\\]\n\n(c)  Given that $B\\subset E$, compute \n\\[\n\\mathbb{E}\\!\\left[\\Sigma(B)\\mid B\\subset E\\right],\n\\]\nwhere $\\Sigma(B)$ denotes the total length of the $12$ edges of $B$.",
      "solution": "Step 1 –  A correct containment criterion  \nWrite $p=(x_{0},y_{0},z_{0})$ and $q=(a,b,c)$.  \nBecause $B$ is axis–parallel, every vertex $v=(x',y',z')$ of $B$ is obtained by independently choosing\n\\[\nx'\\in\\{x_{0},a\\},\\qquad y'\\in\\{y_{0},b\\},\\qquad z'\\in\\{z_{0},c\\}.\n\\]\n\n($\\Longrightarrow$)  Assume $B\\subset E$.  \nIf, say, $\\lvert a\\rvert>\\lvert x_{0}\\rvert$, then the vertex $v=(a,y_{0},z_{0})$ of $B$ satisfies\n\\[\n\\frac{a^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}>\\frac{x_{0}^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}=1,\n\\]\nbecause the left–hand side increases strictly with $\\lvert x\\rvert $.  Hence $v\\notin E$, contradicting $B\\subset E$.  \nTherefore $\\lvert a\\rvert\\le\\lvert x_{0}\\rvert$.  \nExactly the same argument with the other coordinates yields\n\\[\n\\lvert a\\rvert\\le\\lvert x_{0}\\rvert,\\qquad\n\\lvert b\\rvert\\le\\lvert y_{0}\\rvert,\\qquad\n\\lvert c\\rvert\\le\\lvert z_{0}\\rvert .\n\\tag{1}\n\\]\n\n($\\Longleftarrow$)  Conversely, suppose (1) holds.  \nFor any vertex $v=(x',y',z')$ of $B$ we then have\n\\[\n\\frac{x'^{2}}{9}\\le\\frac{x_{0}^{2}}{9},\\;\n\\frac{y'^{2}}{4}\\le\\frac{y_{0}^{2}}{4},\\;\nz'^{2}\\le z_{0}^{2},\n\\]\nso that\n\\[\n\\frac{x'^{2}}{9}+\\frac{y'^{2}}{4}+z'^{2}\n\\;\\le\\;\n\\frac{x_{0}^{2}}{9}+\\frac{y_{0}^{2}}{4}+z_{0}^{2}\n=\\;1.\n\\]\nThus every vertex of $B$ (and hence all of $B$) lies in $E$; i.e. $B\\subset E$.  \nWe have proved\n\\[\nB\\subset E\\quad\\Longleftrightarrow\\quad \\text{the three inequalities in (1)}.\n\\]\n\nStep 2 –  Probability of containment for fixed $p$  \nFor fixed $p$, the admissible points $q$ form the rectangular box\n\\[\nR(p)=\\{(a,b,c):\\lvert a\\rvert\\le\\lvert x_{0}\\rvert,\\;\n                       \\lvert b\\rvert\\le\\lvert y_{0}\\rvert,\\;\n                       \\lvert c\\rvert\\le\\lvert z_{0}\\rvert\\},\n\\]\nwhose volume is\n\\[\n\\operatorname{Vol}(R(p))=8\\lvert x_{0}y_{0}z_{0}\\rvert .\n\\]\nSince $R(p)\\subset E$, the conditional probability is\n\\[\n\\mathbb{P}(B\\subset E\\mid p)=\n\\frac{\\operatorname{Vol}(R(p))}{\\operatorname{Vol}(E)}\n    =\\frac{8\\lvert x_{0}y_{0}z_{0}\\rvert}{\\tfrac{4}{3}\\pi\\cdot3\\cdot2\\cdot1}\n    =\\frac{\\lvert x_{0}y_{0}z_{0}\\rvert}{\\pi}.\n\\tag{2}\n\\]\n\nStep 3 –  Expectation of $\\lvert x_{0}y_{0}z_{0}\\rvert$  \nBecause $\\theta,\\varphi$ are independent and uniform,\n\\[\nx_{0}=3\\sin\\theta\\cos\\varphi,\\quad\ny_{0}=2\\sin\\theta\\sin\\varphi,\\quad\nz_{0}=\\cos\\theta .\n\\]\nHence\n\\[\n\\lvert x_{0}y_{0}z_{0}\\rvert\n      =6\\lvert\\sin^{2}\\theta\\cos\\theta\\sin\\varphi\\cos\\varphi\\rvert .\n\\]\nWith joint density $(2\\pi^{2})^{-1}$,\n\\[\n\\begin{aligned}\n\\mathbb{E}\\bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\\bigr]\n&=\\frac{6}{2\\pi^{2}}\n  \\int_{0}^{2\\pi}\\lvert\\sin\\varphi\\cos\\varphi\\rvert\\,d\\varphi\n  \\int_{0}^{\\pi}\\lvert\\sin^{2}\\theta\\cos\\theta\\rvert\\,d\\theta  \\\\\n&=\\frac{6}{2\\pi^{2}}\\cdot2\\cdot\\frac{2}{3}\n  \\;=\\;\\frac{4}{\\pi^{2}} .\n\\end{aligned}\n\\]\n\nStep 4 –  Answer to part (a)  \nUsing (2),\n\\[\nP=\\mathbb{E}_{p}\\bigl[\\mathbb{P}(B\\subset E\\mid p)\\bigr]\n  =\\frac{1}{\\pi}\\,\\mathbb{E}\\bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\\bigr]\n  =\\frac{4}{\\pi^{3}} .\n\\]\n\nStep 5 –  Conditional expectation of $\\operatorname{Vol}(B)$  \nFor fixed $p$ and $B\\subset E$, $(a,b,c)$ is uniform on $R(p)$, so\n\\[\n\\mathbb{E}[\\,\\lvert x_{0}-a\\rvert\\mid p,B\\subset E]=\\lvert x_{0}\\rvert,\\quad\n\\mathbb{E}[\\,\\lvert y_{0}-b\\rvert\\mid p,B\\subset E]=\\lvert y_{0}\\rvert,\\quad\n\\mathbb{E}[\\,\\lvert z_{0}-c\\rvert\\mid p,B\\subset E]=\\lvert z_{0}\\rvert .\n\\]\nThus\n\\[\n\\mathbb{E}\\bigl[\\operatorname{Vol}(B)\\mid p,B\\subset E\\bigr]\n      =\\lvert x_{0}y_{0}z_{0}\\rvert .\n\\tag{3}\n\\]\n\nUnconditioning and using (2)–(3),\n\\[\n\\begin{aligned}\n\\mathbb{E}\\bigl[\\operatorname{Vol}(B)\\,1_{B\\subset E}\\bigr]\n      &=\\mathbb{E}_{p}\\!\\left[\n        \\mathbb{P}(B\\subset E\\mid p)\\,\n        \\lvert x_{0}y_{0}z_{0}\\rvert\n        \\right]\\\\\n      &=\\frac{1}{\\pi}\\,\n        \\mathbb{E}\\bigl[(x_{0}y_{0}z_{0})^{2}\\bigr].\n\\end{aligned}\n\\]\nCompute the second moment:\n\\[\n(x_{0}y_{0}z_{0})^{2}\n      =36\\sin^{4}\\theta\\cos^{2}\\theta\\cos^{2}\\varphi\\sin^{2}\\varphi .\n\\]\nUsing\n\\[\n\\int_{0}^{\\pi}\\sin^{4}\\theta\\cos^{2}\\theta\\,d\\theta=\\frac{\\pi}{16},\\qquad\n\\int_{0}^{2\\pi}\\cos^{2}\\varphi\\sin^{2}\\varphi\\,d\\varphi=\\frac{\\pi}{4},\n\\]\nwe find\n\\[\n\\mathbb{E}\\bigl[(x_{0}y_{0}z_{0})^{2}\\bigr]\n    =\\frac{1}{2\\pi^{2}}\\cdot36\\cdot\\frac{\\pi}{16}\\cdot\\frac{\\pi}{4}\n    =\\frac{9}{32}.\n\\]\nTherefore\n\\[\n\\mathbb{E}\\!\\left[\\operatorname{Vol}(B)\\mid B\\subset E\\right]\n   =\\frac{\\dfrac{9}{32\\pi}}{P}\n   =\\frac{9\\pi^{2}}{128}.\n\\]\n\nStep 6 –  Conditional expectation of the total edge length  \nFor fixed $p$,\n\\[\n\\Sigma(B)=4\\bigl(\\lvert x_{0}-a\\rvert+\\lvert y_{0}-b\\rvert+\\lvert z_{0}-c\\rvert\\bigr),\n\\]\nso\n\\[\n\\mathbb{E}\\bigl[\\Sigma(B)\\mid p,B\\subset E\\bigr]\n         =4\\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr).\n\\]\nHence\n\\[\n\\mathbb{E}\\bigl[\\Sigma(B)\\,1_{B\\subset E}\\bigr]\n   =\\frac{4}{\\pi}\\;\n     \\mathbb{E}\n        \\Bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\n              \\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr)\\Bigr].\n\\]\n\nBecause of symmetry,\n\\[\n\\mathbb{E}\\bigl[X^{2}YZ\\bigr]\n      =6/\\pi^{2},\\quad\n\\mathbb{E}\\bigl[XY^{2}Z\\bigr]\n      =4/\\pi^{2},\\quad\n\\mathbb{E}\\bigl[XYZ^{2}\\bigr]\n      =\\frac{3}{4\\pi},\n\\]\nwith $X=\\lvert x_{0}\\rvert$, $Y=\\lvert y_{0}\\rvert$, $Z=\\lvert z_{0}\\rvert$.  \nAdding these expectations gives\n\\[\n\\mathbb{E}\n   \\Bigl[\\lvert x_{0}y_{0}z_{0}\\rvert\n         \\bigl(\\lvert x_{0}\\rvert+\\lvert y_{0}\\rvert+\\lvert z_{0}\\rvert\\bigr)\\Bigr]\n   =\\frac{10}{\\pi^{2}}+\\frac{3}{4\\pi}.\n\\]\nConsequently,\n\\[\n\\mathbb{E}\\!\\left[\\Sigma(B)\\mid B\\subset E\\right]\n  =\\frac{\\dfrac{4}{\\pi}\\Bigl(\\dfrac{10}{\\pi^{2}}+\\dfrac{3}{4\\pi}\\Bigr)}{P}\n  =10+\\frac{3\\pi}{4}.\n\\]\n\n\\bigskip\nFinal answers:\n\\[\n\\boxed{P=\\dfrac{4}{\\pi^{3}}},\\qquad\n\\boxed{\\mathbb{E}[\\operatorname{Vol}(B)\\mid B\\subset E]=\\dfrac{9\\pi^{2}}{128}},\\qquad\n\\boxed{\\mathbb{E}[\\Sigma(B)\\mid B\\subset E]=10+\\dfrac{3\\pi}{4}}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.538563",
        "was_fixed": false,
        "difficulty_analysis": "• Dimension raised from 2 D to 3 D: containment must be checked for a rectangular parallelepiped inside a solid ellipsoid.  \n• Three independent angular and Cartesian integrations replace a single one; Beta–functions, trigonometric identities and multi–variable expectations are required.  \n• Parts (b) and (c) involve conditional expectations that demand careful separation of the randomness of p and q, higher mixed moments (second and third), and deft use of symmetry.  \n• The final answers are non-elementary constants (e.g. 9π²/128), showing that mere pattern spotting from the planar case is impossible.  \n• The solution chain uses geometry, measure theory, probability, multivariate calculus and special-function identities—considerably deeper than in the original or the prior kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}