Grinstead-Snell-Probability · Problem 11

Continuous Probability Densities

Mathematics · bertrands-paradox

Problem

Here is another way to pick a chord at random on the circle of unit radius. Imagine that we have a card table whose sides are of length 100. We place coordinate axes on the table in such a way that each side of the table is parallel to one of the axes, and so that the center of the table is the origin. We now place a circle of unit radius on the table so that the center of the circle is the origin. Now pick out a point (x0,y0)(x_0, y_0) at random in the square, and an angle θ\theta at random in the interval (−π/2,π/2)(-\pi/2, \pi/2). Let m=tan⁡θm = \tan\theta. Then the equation of the line passing through (x0,y0)(x_0, y_0) with slope mm is y=y0+m(x−x0),y = y_0 + m(x - x_0), and the distance of this line from the center of the circle (i.e., the origin) is d=∣y0−mx0m2+1∣.d = \left|\frac{y_0 - mx_0}{\sqrt{m^2+1}}\right|. We can use this distance formula to check whether the line intersects the circle (i.e., whether d<1d < 1). If so, we consider the resulting chord a random chord. This describes an experiment of dropping a long straw at random on a table on which a circle is drawn. Write a program to simulate this experiment 10000 times and estimate the probability that the length of the chord is greater than 3\sqrt{3}. How does your estimate compare with the results of Example 2.6?

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