Download Buffon`s Needle Experiment (More than just another dead white

Buffon’s Needle Experiment
(More than just another dead white French male…)
What’s wrong with this picture?
Andrew Caglieris
Mathematics
Department,
Peddie School,
Hightstown, NJ
[email protected]
© January 22nd, 2013
Andrew Caglieris, NCSSM 2013 Teaching Contemporary Mathematics Conference
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Overview of the Talk
 Why Buffon’s Needle Experiment?
(Isn’t that just about a very familiar method for estimating π?)
 Can we have some History please?
Context Matters!
(… my wife is the last bastion of European History at Peddie)
 Conducting the Experiment in a Statistics class setting to arrive
at a confidence interval
( Yes, an opportunity to do a hands on experiment in a math class…)
 Using Calculus Methods to arrive at an exact value for the
parameter being estimated in the experiment
(Yes, an opportunity to use Calculus methods to support a result
obtained in an experimental setting…)
 Building a Mathematical Model for the Distribution of Sample
Proportions
(Yes, we even get to do some mathematical modeling…)
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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Why Buffon’s Needle Experiment to get at
the notion of a distribution of sample
proportions, and confidence intervals?
Why not use Coins? Hershey’s Kisses? Thumb tacks? Beads?
Opportunity to connect Geometric Probability, Calculus, and
Statistics to explore a distribution of sample proportions with a
measure of center probably unknown to most students but yet a
value that we can subsequently prove via Calculus methods.
Opportunity to incorporate a Progression of Activities (ranging
from hands on trials to use of a computer simulation) in
foreshadowing the concept of a confidence interval in estimating
the value of an unknown parameter.
Opportunity to utilize Calculus methods to compare the exact
value with the estimates obtained through examining the
distribution of sample proportions.
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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Can we have some History please?
Context Matters!
Born on Sept. 7th, 1707, in Montbard on the Côte d'Or, in France.
At the age of ten his mother inherited a large sum of money which
allowed George-louis Leclerc to become the lord of Buffon and Montbard.
Georges-louis Leclerc, was educated at the Jesuit College of Godrans in
Dijon until 1723. Despite showing a talent and interest for
mathematics, he followed his father's wishes and began to study law in
1723. At the age of 20 Buffon discovered the binomial theorem.
He carried out his probability experiment by throwing sticks over his
shoulder onto a tiled floor and counting the number of times the
sticks fell across the lines between the tiles. (circa 1777)
Died: 16 April 1788 in Paris, France
Source:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Buffon.html
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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Conducting the Experiment in a
Statistics class setting to arrive at a
confidence interval
(No Needles involved and toothpicks should not be used for their regular purpose)
Seems rather easy…maybe not…
Some questions to consider:
1. How do we drop the toothpicks? (Independence Assumption)
2. How many toothpicks should you include in your sample? (10,
100, 1000?) (Large Enough Sample Size Assumption)
….You mean assumptions and related conditions matter?
How can we address the above theoretical and practical
concerns?
3. Ready for data collection stage using hands on trials
4. Examine distribution of sample proportions (Post-It distribution)
5. Collect more data using an online simulation (Java Applet) or
from a simulation developed by Tim Corica
6. Use concept of a confidence interval to estimate the parameter
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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So what does this look like after
performing a MonteCarlo
Simulation?
Dot Plot
Collection 1
0.3
0.4
mean Proportion_of_hits_100  = 0.6382
0.5
0.6
Proportion_of_hits_100
0.7
0.8
0.9
Dot Plot
Collection 1
0.3
0.4
0.5
0.6
Proportion_of_hits_1000
0.7
0.8
0.9
mean Proportion_of_hits_1000  = 0.63678
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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Using Calculus Methods to arrive at an
exact value for the parameter being
estimated in the experiment
Nice Analysis Presented by George Reese
Also, Buffon’s Needle Experiment is
referred to in the:
Numb3rs Season 2:
Judgement Call Episode
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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Building a Mathematical Model for the
Distribution of Sample Proportions
Bernoulli Trials
1. Independent Trials
2. Two possible outcomes on each trial
3. Fixed probability of success, failure on each trial
Binomial Model
Fixed number of trials: Mean=
S .D=. σ=
µ= np=
npq=
2n
π
 2  2 
n  1 − 
 π  π 
Provided Assumptions hold and Conditions are met we

 2  2  
can use a Normal Model to describe the distribution of 
 1 −  
2
π
π
N  ,  
sample proportions. Large enough sample size?
π

n


Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference


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Going Beyond the Current
Problem…
Some Interesting Avenues to be Explored…
What if the distance between the lines on the grid is less
than the length of the toothpick?
What if the distance between the lines on the grid is greater
than the length of the toothpick?
Also well-known is another problem involving Geometric
Probability and attributed to Buffon. Buffon’ s coin problem
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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It Always Comes Down to Pie!
Nice simulation to arrive at an approximation for π
developed by Tim Corica
Andrew Caglieris, NCSSM 2013 Teaching
Contemporary Mathematics Conference
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