Probability of a small square existing in a larger square without intersecting the larger squares diagonals

by Davis   Last Updated August 14, 2019 06:20 AM

Extra data: The length of the larger squares side is 2 units and the smaller one is 0.5 units The smaller square can have any orientation The smaller square fully lies in the larger one.

My attempts at a solution I first treated this as a generic probab question and tried solving it graphically with diagrams for the extreme cases and then using the fractions of areas to get the answer but then I realized that there were infinite cases where the squares were in the correct position and a larger(?) infinity where the smaller square intersected the diagonals. If this were true, then the required probability tends to zero but this implies that the probability that the smaller square intersects the diagonals tends to one which is intuitively wrong. I then tried solving parametrically, taking one random point in the large square, finding the corresponding possible points for the smaller square with respect to the first point and theta(Angle with the horizontal) then with the inequalities we get comparing these points with the line equations of the diagonals, we can get the probability by dividing the possible theta values by 360. However, in this case, when the first point is selected close to the squares edge, we get some cases that don't even fit in the sample space with the necessary conditions.

I'm on mobile so please excuse formatting or spelling errors

Tags : probability


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