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points or pigeons in square? (Posted on 2019-08-05) Difficulty: 2 of 5
Given 5 points inside a square of side 10 cm, what is the probability that at least two of the points are within 8cm of each other?

No Solution Yet Submitted by Danish Ahmed Khan    
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computer soln | Comment 1 of 8
For me this is easier by simulation, although to do it analytically is 
tractable and would be fun. I get the probability to be about 0.442. 

I wonder if this problem is equivalent to asking this question: "If you connect two random points and do this 10 times, what are the chances that two or more distances will be >8?" I think it is the same. 

(Later....) I answered my own q. - The answer is no. The prob. that 2 or more of 10 random line segments are longer than 8 cm is about 0.455. Distinctly larger probability. This makes the analytic path a bit more complex...

 exp = 6 below means 10^6 random cases.

 exp prob            5  0.442479998    

 exp prob            6  0.441929996    

 exp prob            7  0.442253202    


        program sq5

        implicit none

        integer seed,pa(2,10),flag,i,j,exp,big,cntng

        real x(5),y(5),dx2,dy2,d,prob,xbig

        data pa/1,2,1,3,1,4,1,5,2,3,2,4,2,5,3,4,3,5,4,5/

        seed=time8()

        call srand(seed)

1       print*,'exp?'

        read*,exp

        big=10**exp

        xbig=big

        if(exp.eq.0)stop

        cntng=0

           do i=1,big

                do j=1,5

                x(j)=10*rand()

                y(j)=10*rand()

                enddo

           flag=0

                do j=1,10

                dx2=(x(pa(1,j))-x(pa(2,j)))**2

                dy2=(y(pa(1,j))-y(pa(2,j)))**2

                d=dx2+dy2

                   if(d.ge.64)then

                   if(flag.eq.1)go to 2

                   flag=1

                   endif

                enddo

           cntng=cntng+1

2          enddo

        prob=(xbig-cntng)/xbig

        print*,'exp prob ',exp,prob

        go to 1

        end


Edited on August 5, 2019, 12:22 pm
  Posted by Steven Lord on 2019-08-05 11:04:11

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