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  "body": "\n## Summary\n\nThe Boolean model is a random set in $\\R^d$.\nIt is the union of independent random shapes, the grains, placed at the points of a [[Poisson point process|0MTNR28K60]], the germs.\nIt is the basic germ-grain model of [[stochastic geometry|2J9GB49FHA]].\nMany of its mean values have closed formulas.\nFor example, a given point is covered with probability $1 - e^{-\\lambda E[V_d(K)]}$, where $\\lambda$ is the intensity and $E[V_d(K)]$ the mean grain volume.\nIt is used to model porous media, coverage of wireless networks and continuum [[percolation|BH7YPXCYDK]].\n\n## Explanations\n\n### L1 human\n\nThrow points at random over the plane, with no point influencing any other.\nAround each point draw a shape, such as a disc whose radius is also chosen at random.\nThe Boolean model is the union of all these shapes, a random pattern of overlapping blobs.\nIf all discs have radius $r$ and there are on average $\\lambda$ centres per unit area, then each fixed point of the plane is uncovered with probability $e^{-\\lambda \\pi r^2}$.\nWith $\\lambda \\pi r^2 = 1$, about $63\\%$ of the plane is covered.\n",
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