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Contains The Origin

Problem

Two points, A and B, are selected at random on a co-ordinate plane. Lines are drawn through each of the points that are parallel to both the x and y axes so as to form a rectangular region.


Three points, X, Y, and Z, are selected at random on the same plane so as to form a triangle.

Which shape is most likely to contain the origin?


Solution

Given A, the only way that the rectangle will contain the origin is if B is found in the diagonally opposite quadrant. That is, the probability that the rectangle contains the origin is 1/4.

Given X and Y, we can draw lines XO and YO so as to form four regions.


It is clear that the triangle will contain the origin if Z is chosen within the shaded region, and as X and Y are determined randomly, we shall assume that the area of the shaded region will be uniformly distributed from 0 to 1/2 of the plane. Therefore the expected area will be 1/4.

Hence the probability that each shape contains the origin is 1/4.

Can we assume that the area of the shaded region is uniformly distributed?
What assumptions must be made about the way that X and Y are chosen?
What is the probability that a quadrilateral, formed from the four randomly selected points A, B, C, and D, contains the origin?

Problem ID: 237 (02 Aug 2005)     Difficulty: 4 Star

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