Buffon's Coin Experiment
Buffon's Coin Experiment |
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Simulation of Buffon's coin experiment
Buffon's coin experiment is a very old and famous one. The experiment consists of dropping a coin randomly on a floor covered with identically shaped tiles. The event of interest is that the coin crosses a crack between tiles. We will model Buffon's coin problem first with square tiles of side length 1. (Assuming the side length is 1 is equivalent to measuring distance in units of side length.)
First, let us define the experiment mathematically. As usual, we will idealize the physical objects by assuming that the coin is a perfect circle with radius r and that the cracks between tiles are line segments. A natural way to describe the outcome of the experiment is to record the center of the coin relative to the center of the tile where the coin happens to fall. More precisely, we will construct coordinate axes so that the tile where the coin falls occupies the square
Now when the coin is tossed, we will denote the center of the coin by (X, Y) in S so that S is our sample space and X and Y are our basic random variables. Finally, we will assume that r < 1/2 so that it is at least possible for the coin to fall inside the square without touching a crack.
Next, we need to define an appropriate probability measure that describes our basic random vector (X, Y). If the coin falls "randomly" on the floor, then it is natural to assume that (X, Y) is uniformly distributed on S. By definition, this means that
The applet for this page is a simulation of Buffon's coin experiment. Buffon's floor and a picture box of the sample space are shown. Each time you run the simulation, the coin falls on the floor, the outcome is plotted as a red dot in the sample space, and the values of the variables X and Y are recorded. The coin radius r can be varied with a scroll bar from 0.01 to 0.50 in increments of 0.01.
1. Run
Buffon's coin experiment with the default settings. Watch how the
points seem to fill the sample space S in a uniform
manner.
Our main interest is in the event C that the needle crosses a crack. However, it turns out to be easier to describe the complementary event that the needle does not cross a crack.
2. Show
that
Cc = {r - 1/2 < X < 1/2 - r, r - 1/2 < Y < 1/2 - r}
3. Use
the result of exercise 3 to show that
P(Cc) = (1 - 2r)2, P(C) = 1 - (1 - 2r)2.
In the simulation of Buffon's coin experiment, an indicator variable I is used to indicate whether C or its complement occurs on each run. Thus I = 0 if the coin does not cross a crack and I = 1 if the coin does cross a crack. The value of I is recorded on each update, and the density and empirical density functions are shown in the last graph and table. The boundary square between C and Cc is shown in blue in the scatterplot. Thus, the coin crosses a crack if and only if the outcome falls outside of this square.
4. Change
the radius with the scroll bar and watch how the events C
and Cc change. Run the experiment
with various values of r 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. Solve
Buffon's coin problem with rectangular tiles that have height h
and width w.
6. Solve
Buffon's coin problem with equilateral triangular tiles that have
side length l.
Buffon's Experiments |
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