What Is Buffons Needle Experiment? And What Does It Have to Do With Pi?

Pi is one of those constants in the mathematical globe. And because of that, information technology often appears in nature, especially whatsoever that involves curves and ​spirals. Only it would be as well piece of cake to derive pi from those very things - they are ordered, unchanging, predictable. But what if pi tin can be derived from a serial of improbable, unpredictable actions? "Suppose we have a floor made of parallel strips of forest, each the same width, and nosotros drop a needle onto the floor. What is the probability that the needle volition lie beyond a line between two strips?" ​Georges-Louis Leclerc - Comte de Buffon, 'Histoire Naturelle, Générale et Particulière' (1777) ​Children all over the world (and no doubt many grownups, too) play at "lines and squares", attempting to avoid stepping on the joints or cracks between the panels of pavement in the sidewalk. In this note, we will explore a randomized, mathematical version of the game. The idea was first raised past Georges Louis Leclerc, Comte de Buffon in his paper, Sur le jeu de franc-carreau, published in 1777. Buffon first considered the case of a small-scale money ("united nations ecu") thrown at random on a floor tiled with congruent squares. Information technology was apparently then all the rage to bet every bit to whether the coin would state entirely within the boundaries of a single tile ("à franc-carreau"), or would country across one of the boundaries between two adjacent tiles. Buffon was apparently the first to realize that the fair odds for such a wager could readily exist determined by noting that the money would land entirely within a square tile whenever the center of the coin landed within a smaller foursquare, whose side was equal to the side of a floor tile less the diameter of the coin. Conspicuously, the probability that the money lands wholly within a single tile is simply the ratio of the area of the tile to the area of the square contained within the dashed lines, as the center of the coin is equally likely to land in any two regions of equal area, or, more generally, the probability that the centre of the money lands in one of two regions equals the ratio of the areas of the 2 regions. This seems to accept been the first of the written report of "geometric probability," where probabilities are determined by comparing of measurements, rather than past identifying and counting alternative, equally probable discreet events, such as specific easily of cards, or rolls of dice. Buffon then raises the question of a more than interesting instance -- suppose one throws, not a circular object, merely an object of a more complex shape, such as a square, a needle, or a "baguette" (a rod or stick). He treats in detail the famous "Needle Problem": Suppose a needle is thrown at random on a flooring marked with equidistant parallel lines. What is the probability that the needle will country on one of the lines? Ever helpful, Buffon points out that "On peut jouer ce jeu sur un damier avec une aiguille à coudre ou une épingle sans tête." (Y'all can play this game on a checkerboard with a sewing-needle or a pin without a head.) It is instructive to outset with a simpler problem: what is the probability that a disk of diameter d volition state on one of the lines, where the lines are separated past a altitude h? As in the example of square tiles, it can be seen by inspection that the disk will country between the lines whenever the heart of the disk lands in a ring of width h - d. Thus, the probability that a disk of bore d lands betwixt the lines is ( h - d ) / h, and the probability that the disk lands on a line is only d/h. Now consider the example of the needle thrown on the flooring ruled with equidistant parallel lines. For simplicity, suppose that the length of the needle is equal to the spacing, h, between adjacent lines. For a moment, let'southward restrict our attention to those throws when the needle falls perpendicular to the lines on the floor. In this case, the needle acts just like a disk of diameter h as in the previous example -- the probability that the needle crosses a line is h / h = 1. Similary, if the needle falls parallel to the ruled lines, the probability that it crosses a line is zilch. Now consider the case of a needle which falls then as to brand an angle, θ, with the parallel lines on the floor. And then reference to the figure confirms that in this instance, the needle has an effective "summit" of h sin θ, and and so the probability that the needle in this orientation will cross a line is h sin θ / h = sin θ. Of form, this formula includes the special cases noted above where the needle falls perpendicular or parallel to the lines on the flooring. In lodge to make up one's mind the overall probability that the needle lands on a line, nosotros must account for the probability of each possible orientation of the needle, and then employ our formula for the probability that the needle, in that orientation, lands so every bit to cross a line on the flooring. This sort of calculation is an application of the notion of "conditional probability". Buffon discovered that if yous describe a prepare of equally-spaced parallel lines (say, d centimetres apart) and drop sticks on them which are shorter than the spacing (say 50 centimetres long, where fifty is less than d), then the probability of a stick crossing a line is: 2l/ πd​ This means that if yous drib lots of sticks randomly and count how many cross the parallel lines, you can calculate what π is by rearranging the formula: π = 2 fifty due south/cd where s is the number of sticks yous driblet and c is the number that crossed a line. Isn't that remarkable? Buffon had worked out that you tin can calculate π just past dropping a bunch of sticks on a table — no circles required! The problem is known as Buffon'south Needle. As usual in statistics, we desire to take a big sample to go an gauge close to the truthful probability. If I ask five people 'are you left handed?' and y'all ask 500 people the aforementioned, we know that your bigger sample is likely to give u.s.a. the meliorate idea of what proportion of people are left handed in the whole land. The same applies here — to get a proficient sample we need to drop a lot of sticks. Perhaps that'south why information technology took a long time afterward Buffon'south discovery for somebody to actually endeavor it themselves, but in 1901 the Italian mathematician Mario Lazzerini gave it a go. He span effectually and dropped over 3,400 sticks onto the floor, counted up the number that crossed over lines and estimated π to be 3.1415929. We know π to exist 3.1415927…, and so he was correct in the first 6 digits, an error of 0.000006%. That's expert! Suspiciously good in fact, and so improbable that it's more likely he cheated. Simulations have go vastly easier and more than of import at present that we can employ computers to do the legwork for us. I'm sure a dizzy Lazzerini would've appreciated one. Simulations like this, where we employ randomness (or, usually, pseudo-randomness) to requite united states a sample of results, are called Monte Carlo methods, later the European city famous for its casino. Monte Carlo methods are very useful in physics, as we can model a big system with a huge number of possible states with a much smaller, but representative subset. It has of import uses in thermal physics, molecular modelling, astrophysics and weather forecasting. And, as Buffon showed 200 years ago, tin even be used to calculate π.   Ponder this Would the gauge exist more than or less accurate if the needle is shortened or diffuse? What if instead of a needle (which have equidistant length from its centre, creating a circle), nosotros apply a square (which length betwixt its perimeter and middle varies) instead?   Hash out ​Try repeating the experiment in class. Group the students into teams and have them compete with each other in estimating pi.   Further readings

Topics

moorepura1937.blogspot.com

Source: http://ifsa.my/articles/buffons-needle-the-improbability-of-pi

Share this post