What is the principle of the Pufeng experiment? Can anyone explain the Pufeng experiment

In his 1777 book "Experiments in Arithmetic of Probability", he proposed his famous needle-throwing problem, and Pu Feng proposed the use of experimental probability to calculate This experimental method is very simple: find a thin needle with uniform thickness and length d, draw a set of parallel lines spaced with l spacing on a blank piece of paper (for convenience, often take l = d 2), and then throw the needle on the white paper again and again. Throw this repeatedly, counting the number of times the needle intersects with any parallel line, and you can get an approximation.

Because Pufeng himself proved that the probability of a needle intersecting with any parallel line is p = 2l d. Using this formula, an approximation of pi can be obtained using the probabilistic method. In one experiment, he chose l = d 2 and cast the needle 2212 times, where the needle intersected the parallel line 704 times, so that the approximation of pi was 2212 704 = .

When a considerable number of votes are made in the experiment, a more precise value can be obtained.

In 1850, a man named Wolfe gave an approximation of after throwing more than 5,000 times. At present, it is the Italian Lazreni who claims to have achieved the best results with this method. In 1901, he repeated the experiment with 3,408 needle throws, and obtained an approximation that the result was so accurate that many doubted the authenticity of the experiment.

For example, L. Badger of National Weber University in Ogden, Utah, USA, has strongly questioned this.

However, the importance of the Pufeng experiment is not to obtain more accurate values than other methods. The importance of the Pin Throwing problem lies in the fact that it is the first example of a probability problem expressed in geometric form. This method of computation is not only amazing because of its novelty and wonder, but also creates a precedent for using random numbers to deal with deterministic mathematical problems, and is a precursor to using chance methods to solve deterministic calculations.

Buffon needle throwing problem.

In his 1777 book Experiments on Probability Arithmetic, Buffon proposed a method for calculating pi - the random needle throwing method, which is known as the Pufeng needle throwing problem.

The operation of this experimental method is very simple:

1) Take a blank piece of paper and draw many parallel lines with d spacing on it;

2) Take a needle with a length of l(l3) and calculate the probability of the needle intersecting the straight line.

The sufficient and necessary conditions for the intersection of the needle and the parallel line are known by the analysis.

Establish a Cartesian coordinate system, and the above conditions will be a curved trapezoidal area surrounded by curves in the coordinate system, which is known by geometric probability.

4) The probability is estimated by statistical experiments.

By (*) i.e.

Throw this repeatedly, counting the number of times the needle intersects with any parallel line, and you can get an approximation. Thus, Pu Feng himself proved that the probability of the needle intersecting any parallel line is p = 2l d. Using this formula, an approximation of pi can be obtained using the probabilistic method.

In one experiment, he chose l = d 2 and cast the needle 2212 times, where the needle intersected the parallel line 704 times, so that the approximation of pi was 2212 704 = . When a considerable number of votes are made in the experiment, a more precise value can be obtained.

Is it the Pufeng needle throwing experiment?

Pufeng needle throwing experiment is a method of calculating pi - random needle throwing method, which is the famous Buffon needle throwing problem. The steps of this method are:

1) Take a blank piece of paper and draw many parallel lines with a distance of d on the upper face.

2) Take a piece of length l (row book disturbance l3) to calculate the probability of the needle intersecting the straight line, this probability is p=2l ( d) is pi.