The Probability of a Crack Crossing
The Probability of a Crack Crossing |
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Simulation of Buffon's needle experiment
In Buffon's needle experiment, our main interest is in the event C that the coin crosses a crack.
1. Use trigonometry to show that C can be written in terms of
the basic angle and distance variables as follows:
2. Use calculus to show that area(C) = 2L and therefore
3. Find the probability that the needle does not cross a crack.
In the simulation of Buffon's needle experiment, an indicator variable I is used to indicate whether C or its complement occurs on each run. Thus, I = 1 if the needle crosses a crack and I = 0 if the needle does not cross a crack. The value of I is recorded on each update, and the density and empirical density functions of I are shown in the third graph and the third table.
The curves
y = (L / 2)sin(x), y = 1 - (L / 2)sin(x)are shown in blue in the scatterplot, and hence event C is the union of the regions below the lower curve and above the upper curve. Thus, the needle crosses a crack precisely when a point falls in this region.
4. Vary the needle length L with the scroll bar and watch how
the events C and Cc change. Run the experiment
with various values of L and compare the physical experiment with
the points in the scatterplot. Note the apparent convergence of relative
frequency of C to the probability of C.
The convergence of the relative frequency of an event (as the experiment is repeated) to the probability of the event is a special case of the law of large numbers.
5. Find the probabilities of the following events in Buffon's needle
experiment. In each case, sketch the event as a subset of the sample space.
Buffon's Experiments |
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