7.7 Simulating Buffon's needle experiment ^{11} Recall Buffon's needle
experiment, the famous early experiment involving geometrical
probability concepts, from Section 3.3. 1. Assuming that the length of
the needle, 1, does not exceed the spacing, d, between parallel lines (I
d), the probability that a randomly thrown needle intersects a line
was shown to be
Thus, by conducting Buffon's experiment many times, one can obtain an estimate for, say , through Suppose that we wish to perform these experiments by using computer simulation. After all, it may take a large amount of time to toss a needle, say, a few thousand times, although this has not been sufficient to discourage several people in the past from doing just that (up to 10,000 consecutive tosses have been reported). Consider the following three alternative methods for simulating Buffon's needle experiment. (For each execution of the experiment, new random numbers must be drawn.) Method 1 1. Obtain a random value for X, say x, from the uniform distribution on the interval [0, d] (i.e., x = r_{1}d). 2. Obtain a random value for S, say s, the sine of the angle which the needle forms with the set of parallel lines (see Figure 3.17) from the uniform distribution on [0, 1] (i.e., s = r_{2}). 3. if x 1/2 Is or if x d - 1/2 ls, an intersection has occurred. Method 2 1. Obtain a random value y_{1} for Y_{1}, from the uniform distribution on the interval [0, d/2] (i.e., y_{1} = r_{1}d/2). Let (0, y_{1}) represent the midpoint of the needle. 2. Obtain a random value, x_{2}, for X_{2}, from the uniform distribution on the interval [-l/2, +l/2] (i.e., X_{2} = r_{2}l - 1/2). 3. Similarly, obtain a random value, y_{2}, for Y_{2}, using y_{2} = r_{3}l - l/2. [(x_{2}, y_{2} + y_{1}) will be the coordinates of some point lying on the needle.] 4. If (x^{2}_{2} + y^{2}_{2})1/2 1/2 l, go to Step 5. Otherwise, return to Step 2 and continue 2 2 1/2l until a set of values (x_{2}, y_{2}) has been obtained that satisfies the foregoing test. 5. if
an intersection has occurred. Method 3 1. Obtain a random value, x_{1}, for X_{1} from the uniform distribution on the interval (0, d] (i.e., x_{1} = r_{1}d). 2. Similarly, obtain a random value, y_{1}, for Y_{1}, using y_{1} = r_{2}d. Let (x_{1}, y_{1}) represent the midpoint of the needle. 3. Similarly, obtain random values (x_{2}, y_{2}), for (X_{2}, Y_{2}) using x_{2} = r_{3}d and y_{2} = r_{4}d. 4. Let the end points of the needle lie on the line joining (x_{1}, y_{1}) with (x_{2}, y_{2}) at a distance 1/2 from (x_{1}, y_{1}) in each direction. 5. If y_{1} 1/2 l sin
or y_{1} d - 1/2 l sin
where
an intersection has occurred. a. Provide a physical interpretation of each one of the three methods by drawing figures that show schematically what each method does. b. Which of the foregoing three methods are correct, which are incorrect, and why? Hint: Two of the methods are incorrect. |