Geometric Probability in Multiple DimensionsI understand how Geometric Probability works in 1, 2, and 3 dimensions, but is it possible to do these problems in, say, 5 dimensions? For example,Five friends are to show up at a party from 1:00 to 2:00 and are to stay for 6 minutes each. What is the probability that there exists a point in time such that all of the 5 friends meet?

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Izabelle Frost · Accepted Answer

Step 1Consider the simpler problem of 2 people. Chart on the x axis the time of stay of first person. Chart on the y axis the time of stay of second person. The cartesian product of these intervals together form a rectangle in             R        2  Question: When do the two meet, and what does this mean geometrically? It&#039;s easy to see that whenever this rectangle intersects the   x  =  y line, then the two people meet (or have an overlapping time), or else if the rectangle is fully below   x  =  y, or fully above they don&#039;t meet.Extending to higher dimensions: When formulated this way, it should be the case that in             R        n  , when the rectangle formed by the cartesian product of the intervals of stay of different people intersects the line       x    1    =      x    2    =  ⋯  =      x    n  , (or in parametric form,   λ  [  1  ,  1  ,  …  ,  1  ]    ∀  λ) then, all the people have some overlapping time of stay.Step 2Probability of overlap: The probability is now the measure of the union of all the rectangles (or hypercubes in             R        n  ) intersecting with the line (      x    1    =      x    2    =  …) vs the measure of the entire space under consideration, say under normalization,   [  0  ,  1  ]  ×  [  0  ,  1  ]  ×  …  [  0  ,  1  ]

Five friends are to show up at a party from 1:00 to 2:00 and are to stay for 6 minutes each. What is the probability that there exists a point in time such that all of the 5 friends meet?

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