synonymBufeng needle(Bufeng needle) generally refers to Pufeng needle problem
In the 18th century,Buffon Put forward the following question: Suppose we have a floor paved with parallel and equidistant wood grains (as shown in the overview diagram), throw a needle whose length is smaller than the distance between the wood grains at random, and calculate the probability of intersection between the needle and one of the wood grains.And with this probability,BuffonA method for calculating Pi - random needle throwing method is proposed.This is the Pufeng needle problem.
French mathematicianBuffon(1707-1788) was the first to design the injection test.
The steps of this method are:
1) Take a piece of white paper and draw many lines with spacing of a on itParallel line。
2) Take a needle with a length of l (l ≤ a), throw it on the paper with parallel straight lines for n times at random, and observe the number of times the needle intersects the straight line, which is recorded as m.
3) Calculate the probability that the needle intersects the straight line
French mathematician in the 18th centuryBuffonThe "needle projection problem" proposed by Bufon was recorded in his book published in 1777: "a group of spaces a are drawn on the planeParallel line, throw a needle with a length of l (l ≤ a) on the plane at will, and calculate the probability of intersection between the needle and any of the parallel lines. "
Buffon himself proved that the probability is:
(where π is pi)
Because it is related to π, people thought of using the needle throwing test to estimate the value of pi.
Buffon was surprised to find that the ratio of the times of favorable throwing and unfavorable throwing is an expression containing π. If the length of the needle is equal to a/2, then the probability of throwing is 1/π. The more times of throwing, the more accurate the value of π can be obtained.
experimental data
The following formula is used to obtain pi by probability methodApproximate valueSome information about.
In 1901,ItalyMathematician Lazrini claimed to have carried out many needle injection tests, with 3408 needles per time, on averageNumber of intersectionsFor 1808 times, the given value of π is 3.1415929 - accurate to 6 decimal places.However, whether or not Lazrini actually threw an injection, his experiment was carried out in the United StatesUtahOgdenL Bajie of National Weber UniversityCalculus, probability and other wide range and channels to find π, which is really surprising!
BuffonThe needle throwing experiment is the first example to express the probability problem in geometric form. It is the first time for him to use randomExperimental treatmentcertaintyMathematical problems, forprobability theoryThe development of has played a certain role in promoting.[1]
prove
Certificate 1:Find an iron wire and bend it into a circle so that its diameter is exactly equal toParallel lineDistance d between.It can be imagined that for such a circle, no matter how it is dropped, it will have two intersections with the parallel line.Therefore, if the circle is dropped n times, the total number of intersecting points must be 2n.Imagine straightening the circle into oneIs longπ d wire.Obviously, it is more complicated for such wire to intersect with parallel lines when it is dropped than a circle. There may be 4 intersections, 3 intersections, 2 intersections, 1 intersection, or even none.Since the length of the circle and the straight line are both π d, according to the principle of equal opportunity, when they are thrown more times and equal, the total number of intersection points between the two and the parallel line group is expected to be the same.That is to say, when the iron wire with the length of π d is dropped n times, the total number of intersections with the parallel line should be about 2n.
Turn to the case where the length of the wire is l.When the number of tosses n increases, this wire followsParallel lineThe maximum total number of intersections m should be proportional to the length l, so: m=kl, where k isScale factor。
In order to find k, note the special case when l=π d, where m=2n.So we get
。
The formula before substitution is as follows:
If this conclusion is extended to l=a/2, then there is only one intersection point at most, and the ratio of m to n is the probability of intersection of needle and straight line.However, this proof is less rigorous. For example, circle and straight line are expected to be equal, and the intersection point of iron wire and parallel line is proportional.Next, we use probability theory and calculus to provide rigorous proof.
Proof II: Because the needle injection to the desktop is random, useTwo-dimensional random variable(X, Y) to determine its specific position on the table.Let X represent the distance between the midpoint of the needle and the parallel line, and Y represent the included angle between the needle and the parallel line. If
Like the needle throwing experiment, we use the probability obtained from the probability experiment to estimate a quantity of interest. This method is called the Monte Carlo method.When this kind of model contains uncertain random factors, it is usually more difficult to analyze than the deterministic model.Some models are difficult to make quantitative analysis and cannot obtain analytical results, or although there are analytical results, the calculation cost is too high to use.In this case, the Monte Carlo method can be considered,Monte Carlo method Is onthe Second World WarDuring this period, it rose and developed with the birth of computers.This method is used inApplied Physics、atomic energy、Solid state physics, chemistryecology, sociology andeconomic behaviorAnd other fields.
Besides, name 3 at randomPositive numberThe probability P that these three positive numbers can form an obtuse triangle is also related to π, and the probability is (π - 2)/4. The proof is as follows:
Let these three positive numbers be x, y, z, let's say x ≤ y ≤ z, for each determined z, it must meet the requirements of x+y>z, x ^ 2+y ^ 2<z ^ 2, and it is easy to prove that these two formulas can form an obtuse triangle with these three positive numbers as the side lengthnecessary and sufficient condition, usinglinear programmingIt can be known thatFeasible regionIs a straight line x+y=z and a circle x ^ 2+y ^ 2=z ^ 2;The total feasible region is a side length of zsquare, then the probability of enclosing an obtuse triangle P=S arch/S square=(π z ^ 2/4-z ^ 2/2)/z ^ 2=(π - 2)/4. Because for each z, the probability is (π - 2)/4, so for any positive number x, y, z, there is P=(π - 2)/4, and the proposition is proved.
In order to estimate the value of π, we need to estimate its probability through experiments. This process can be implemented by computer programming. In fact, x+y>z, x ^ 2+y ^ 2<z ^ 2 is equivalent to (x+y-z) (x ^ 2+y ^ 2-z ^ 2)<0, so we just need to check whether this formula is true.If m timesRandom test, there are n times to meet the equation, when m is enoughLarge time, n/m approaches to (π - 2)/4, let n/m=(π - 2)/4, and solve π=4n/m+2 to estimate the value of π.
It is worth noting the method adopted here: design an appropriate experiment, whose probability is related to a quantity we are interested in (such as π), and then use the test results to estimate this quantity. With the development of modern technologies such as computers, this method has developed into a widely used Monte Carlo method.
Monte Carlo method is the basis of computer simulation. Its name comes from the world famousCasino——MonacoOfMonte Carlo Its history originated from French scientists in 1777Buffon A method for calculating the circumference π, the random needle casting method, is proposed, that is, the famous Pufeng needle casting problem.
The basic idea of the Monte Carlo method is to first establish aprobability modelAnd make the solution of the problem exactly the parameter of the model or other relevant characteristic quantity. Then, by simulating a statistical test, that is, multiple random sampling tests (determine m and n), statistics of the occurrence of an eventpercentage。As long as the number of tests is large, the percentage is close to the probability of the event. This is actually the statistical definition of probability.Using the established probability modelTo make a requestEstimated parameters.Monte Carlo methodBelongs to the branch of experimental mathematics.
arbitrarilyCurved trapezoidThe approximate area is the area of the pond. What should we do?measuring methodAs follows: It is assumed that the pond is located in a rectangular farmland of known area.As shown in Figure 1: Throw stones at this farmland randomly to make them fall into the farmland.The stones thrown into the farmland may or may not be splashed with water. It is estimated that the amount of stones "splashed with water" accounts for the percentage of the total amount of stones.Imagine how to use this estimated percentage to approximate the pond area?
