Mathematics in Terrains

8/11/2019 Mathematics in Terrains

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Some experiments in probability theory - A prelude

to mathematics on terrains

Gopikrishnan C R

School of MathematicsIndian Institute of Science Education and ResearchThiruvananthapuram

September 18, 2014

1 / 15 Gopikrishnan C R Math.On.Terrains
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Birthday Problem

ProblemWhat is the probability that two persons in a group ofn peoplehave their bithdays on the same date and month?

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Birthday Problem


What is the sample space ()?

= {{x1, x2, . . . , xn}|xi is the birthday of the ith person}|| = 365365 365, n times

= 365n

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Birthday Problem




= 365n

What is the event space (E)?

E= {{x1,

x2, . . . ,

xn}|xi=xjfor some i,j}

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Birthday Problem




= 365n

What is the event space (E)?

E= {{x1,

x2, . . . ,

xn}|xi=xjfor some i,j}What we dont require (EC)?EC = {{x1, x2, . . . , xn}|xi=xj for any i,j}

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Probability of same birth dates

|EC| = 365364363 (365n + 1)

= 365!

(365n)!

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f


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|EC| = 365364363 (365n + 1)

= 365!

(365n)!

=

365Pn

|E| = || |EC|

= 365n 365Pn


P b bili f bi h d


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|EC| = 365364363 (365n + 1)

= 365!

(365n)!

=

365Pn

|E| = || |EC|

= 365n 365Pn

Probability ofE=365n 365Pn

365n

= 1365Pn

365n


Wh d hi i ll ?
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What does this convey numerically ?

Final Result

Probability of two persons having the same birth date,p =

1365

Pn365n

Let us do a numerical analysis, how does this function behave.

http://c/Users/user/Desktop/Presentation/birthday.nbhttp://c/Users/user/Desktop/Presentation/birthday.nbhttp://find/http://goback/


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Number of persons p

1 0

10 0.11694820 0.411438

30 0.706316

40 0.89123250 0.970374

60 0.994123

70 0.99916

80 0.999914

90 0.999994

100 1


B ff N dl P bl
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Buffons Needle Problem

Problem

If a short needle, of length l, isdropped on a paper that is ruledwith equally spaced lines of

distance d l, then theprobability that the needle comesto lie in a position where itcrosses one of the lines is exactly

p= 2ld

Figure : Georges-Louis Leclerc,comte de Buffon

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Figure : Needles over a plane ruled by parallel lines



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Let pi is the probability that the needle crosses exactly i timesthe lines.

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Then the expected number of crossings is,

E=p1 + 2p2 + 3p3 + . . .



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E=p1 + 2p2 + 3p3 + . . .

If the needle is short (l d), then p2=p3= = 0

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E=p1 + 2p2 + 3p3 + . . .


There fore in this case we have,

E=p1

That means we are actually searching for the expected

number of crossings of a short needle.

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E=p1 + 2p2 + 3p3 + . . .


There fore in this case we have,

E=p1

That means we are actually searching for the expected

number of crossings of a short needle.Let E(l) denotes the number of crossings that will beproduced by dropping a straight needle of length l.

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Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation

E(l) =

E(x) +

E(y) (1)

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E(

l) =

E(

x) +

E(

y) (1)

Then inductively we shall obtain,

E(nx) =nE(x) (2)

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E(

l) =

E(

x) +

E(

y) (1)


E(nx) =nE(x) (2)

There fore,E(rx) =rE(x) rQ (3)

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E(l) = E(x) +E(y) (1)


E(nx) =nE(x) (2)


Clearly E(x) E(y) ifx y. That is E(x) is a monotonefunction of x.

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E(l) = E(x) +E(y) (1)


E(nx) =nE(x) (2)


Clearly E(x) E(y) ifx y. That is E(x) is a monotonefunction of x.

Solving the above equation we shall obtain,

E(x) =cx (4)

Our aim is to find what is this constant c= E(1)

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Consider a polygonal needle of total length l.

Then the linearity of expectation gives E(l) =cl

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Consider a polygonal needle of total length l.

Then the linearity of expectation gives E(l) =cl

Consider a circular needle C of diameter d. Then the total

length of this needle is d. Irrespective of the position bywhich the needle falls, it will make two crossings.

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Approximatethe circular needle from outside by a polygonalneedle Pn and from inside by Pn. Then clearly,

E(Pn) E(C) E(Pn

) (5)

cL(Pn) 2 cL(Pn)

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Approximatethe circular needle from outside by a polygonalneedle Pn and from inside by Pn. Then clearly,

E(Pn) E(C) E(Pn

) (5)

cL(Pn) 2 cL(Pn)

Taking the limit as n tends to infinity, the polygons becomemore and more same as the circular needle. Thus,

c limn

L(Pn) 2 c limn

L(Pn)

cd 2 cd

c= 2

d (6)

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Final Result

Probability that a short needle crosses a line =

2l

d

There fore if we make M trials and N out of them cross a line, then

N

M

2l

d

2l

dN

M

2lM

dN

http://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://find/


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Final Result

Probability that a short needle crosses a line =

2l

d

There fore if we make M trials and N out of them cross a line, then

N

M

2l

d

2l

dN

M

2lM

dN

Let us implement this experiment.


Polygonal Approximation
http://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://find/


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References
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M. Aigner, K.H. Hofmann, and G.M. Ziegler.Proofs from THE BOOK.Springer, 2010.

L.C Evans.

An introduction to stochastic calculus.W. Feller.AN INTRODUCTION TO PROBABILITY THEORY AND

ITS APPLICATIONS, 2ND ED.

Number v. 2 in Wiley publication in mathematical statistics.Wiley India Pvt. Limited, 2008.

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Mathematics in Terrains