Antimatroid

The a needle lies across a line, while the b needle does not.

In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo method for approximating the number π.

Solution

The problem in more mathematical terms is: Given a needle of length $l$ dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?

Let x be the distance from the center of the needle to the closest line, let θ be the acute angle between the needle and the lines.

The uniform probability density function of x between 0 and t /2 is

${\begin{cases}{\frac {2}{t}}&:\ 0\leq x\leq {\frac {t}{2}}\\0&:{\text{elsewhere.}}\end{cases}}$

The uniform probability density function of θ between 0 and π/2 is

${\begin{cases}{\frac {2}{\pi }}&:\ 0\leq \theta \leq {\frac {\pi }{2}}\\0&:{\text{elsewhere.}}\end{cases}}$

The two random variables, x and θ, are independent, so the joint probability density function is the product

${\begin{cases}{\frac {4}{t\pi }}&:\ 0\leq x\leq {\frac {t}{2}},\ 0\leq \theta \leq {\frac {\pi }{2}}\\0&:{\text{elsewhere.}}\end{cases}}$

The needle crosses a line if

$x\leq {\frac {l}{2}}\sin \theta .$

Now there are two cases.

Case 1: Short needle

This GIF image describes the solution of Buffon's Needle Problem for the "short needle" case

Suppose $l\leq t$.

Integrating the joint probability density function gives the probability that the needle will cross a line:

$P=\int _{\theta =0}^{\frac {\pi }{2}}\int _{x=0}^{(l/2)\sin \theta }{\frac {4}{t\pi }}\,dx\,d\theta ={\frac {2l}{t\pi }}.$

A particularly nice argument for this result can alternatively be given using "Buffon's noodle".

Case 2: Long needle

Suppose $l>t$. In this case, integrating the joint probability density function, we obtain:

$\int _{\theta =0}^{\frac {\pi }{2}}\int _{x=0}^{m(\theta )}{\frac {4}{t\pi }}\,dx\,d\theta ,$

where $m(\theta )$ is the minimum between $(l/2)\sin \theta$ and $t/2$.

Thus, performing the above integration, we see that, when $t<l$, the probability that the needle will cross a line is

${\frac {2l}{t\pi }}-{\frac {2}{t\pi }}\left\{{\sqrt {l^{2}-t^{2}}}+t\sin ^{-1}\left({\frac {t}{l}}\right)\right\}+1$

or

${\frac {2}{\pi }}\cos ^{-1}{\frac {t}{l}}+{\frac {2}{\pi }}{\frac {l}{t}}\left\{1-{\sqrt {1-\left({\frac {t}{l}}\right)^{2}}}\right\}.$

In the second expression, the first term represents the probability of the angle of the needle being such that it will always cross at least one line. The right term represents the probability that, the needle falls at an angle where its position matters, and it crosses the line.

Using elementary calculus

The following solution for the "short needle" case, while equivalent to the one above, has a more visual flavor, and avoids iterated integrals.

We can calculate the probability $P$ as the product of 2 probabilities: $P=P_{1}\cdot P_{2}$, where $P_{1}$ is the probability that the center of the needle falls close enough to a line for the needle to possibly cross it, and $P_{2}$ is the probability that the needle actually crosses the line, given that the center is within reach.

Looking at the illustration in the above section, it is apparent that the needle can cross a line if the center of the needle is within $l/2$ units of either side of the strip. Adding ${\frac {l}{2}}+{\frac {l}{2}}$ from both sides and dividing by the whole width $t$, we obtain $P_{1}={\frac {l}{t}}.$

The red and blue needles are both centered at x. The red one falls within the gray area, contained by an angle of 2θ on each side, so it crosses the vertical line; the blue one does not. The proportion of the circle that is gray is what we integrate as the center x goes from 0 to 1

Now, we assume that the center is within reach of the edge of the strip, and calculate $P_{2}$. To simplify the calculation, we can assume that $l=2$.

Let x and θ be as in the illustration in this section. Placing a needle's center at x, the needle will cross the vertical axis if it falls within a range of 2θ radians, out of π radians of possible orientations. This represents the gray area to the left of x in the figure. For a fixed x, we can express θ as a function of x: $\theta \left(x\right)=\cos ^{-1}\left(x\right)$. Now we can let x move from 0 to 1, and integrate:

$P_{2}=\int _{0}^{1}{\frac {2\theta (x)}{\pi }}\,dx={\frac {2}{\pi }}\int _{0}^{1}\cos ^{-1}(x)\,dx={\frac {2}{\pi }}\cdot 1={\frac {2}{\pi }}.$

Multiplying both results, we obtain $P=P_{1}\cdot P_{2}={\frac {l}{t}}{\frac {2}{\pi }}={\frac {2l}{t\pi }}$, as above.

There is an even more elegant and simple method of calculating the "short needle case". The end of the needle farthest away from any one of the two lines bordering its region must be located within a horizontal (perpendicular to the bordering lines) distance of $l\cos \theta$ (where $\theta$ is the angle between the needle and the horizontal) from this line in order for the needle to cross it. The farthest this end of the needle can move away from this line horizontally in its region is $t$. The probability that the farthest end of the needle is located no more than a distance $l\cos \theta$ away from the line (and thus that the needle crosses the line) out of the total distance $t$ it can move in its region for $0\leq \theta \leq \pi /2$ is given by

$P={\frac {\int _{0}^{\frac {\pi }{2}}l\cos \theta d\theta }{\int _{0}^{\frac {\pi }{2}}td\theta }}={\frac {l}{t}}{\frac {\int _{0}^{\frac {\pi }{2}}\cos \theta d\theta }{\int _{0}^{\frac {\pi }{2}}d\theta }}={\frac {l}{t}}{\frac {1}{\frac {\pi }{2}}}={\frac {2l}{t\pi }}$, as above.

Estimating π

In the first, simpler case above, the formula obtained for the probability $P$ can be rearranged to: $\pi ={\frac {2l}{tP}}$. Thus, if we conduct an experiment to estimate $P$, we will also have an estimate for π.

Suppose we drop n needles and find that h of those needles are crossing lines, so $P$ is approximated by the fraction $h/n$. This leads to the formula:

$\pi \approx {\frac {2l\cdot n}{th}}.$

In 1901, Italian mathematician Mario Lazzarini performed the Buffon's needle experiment. Tossing a needle 3408 times, he obtained the well-known estimate 355/113 for π, which is a very accurate value, differing from π by no more than 3×10⁻⁷. This is an impressive result, but is something of a cheat, as follows.

Lazzarini chose needles whose length was 5/6 of the width of the strips of wood. In this case, the probability that the needles will cross the lines is ${\frac {5}{3\pi }}$. Thus if one were to drop n needles and get x crossings, one would estimate π as

π ≈ 5/3 · n/x.

π is very nearly 355/113; in fact, there is no better rational approximation with fewer than 5 digits in the numerator and denominator. So if one had n and x such that:

355/113 = 5/3 · n/x

or equivalently,

x = 113n/213

one would derive an unexpectedly accurate approximation to π, simply because the fraction 355/113 happens to be so close to the correct value. But this is easily arranged. To do this, one should pick n as a multiple of 213, because then 113n/213 is an integer; one then drops n needles, and hopes for exactly x = 113n/213 successes.

If one drops 213 needles and happens to get 113 successes, then one can triumphantly report an estimate of π accurate to six decimal places. If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate".

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Schroeder, L. (1974). "Buffon's needle problem: An exciting application of many mathematical concepts". Mathematics Teacher, 67 (2), 183–6.

External links

• Buffon's Needle at cut-the-knot
• Math Surprises: Buffon's Noodle at cut-the-knot
• MSTE: Buffon's Needle
• Buffon's Needle Java Applet
• Estimating PI Visualization (Flash)
• Buffon's needle: fun and fundamentals (presentation) at slideshare
• Animations for the Simulation of Buffon's Needle by Yihui Xie using the R package animation
• 3D Physical Animation Java Applet by Jeffrey Ventrella
• Template:Cite web