Re: [Phys-L] From a Math Prof (physics BS major) at my institution ( math challenge)

[Jeffrey Schnick <JSchnick@Anselm.Edu>]

I ran a Monte Carlo Octave function to investigate occurrences of sequences of 2 numbers in a row and, occurrences of round numbers (10,20,30).  In a million sets of 21 rows of five numbers between 1 and 35 inclusive, I got 2784 cases in which there were exactly 2 round numbers and only 133 cases in which there were exactly 2 sequences of 2 numbers in a row.  In 0 out of a million cases there were both 2 or fewer round numbers and 2 or fewer sequences of 2 numbers in a row.  The first set given in this thread met both of these conditions.  In 11 cases out of a million there were both 3 or fewer whole numbers and 3 or fewer sequences of 2 numbers in a row.

It is clear that one can not prove (without looking up the lottery list) which is the student-generated set and which is the lottery-machine-generated set.  Each set is equally probable (I think at about 1 in 1e96).  Based on what John Clement tells us about the correlation between what our intuition tells us is good for student learning and what PER tells us what is good for student learning, Richard Tarara's hypothesis that the students that produced the student-generated set of values would tend to shy away from sequences of numbers in a row, and mine that they would shy away from round numbers could be exactly wrong.  Still: John Denker, if someone offered you a million dollars to correctly pick the student-generated set based on the information in the thread-starting post alone (e.g. without looking up the lottery values on the internet), with the outcome being decided by a coin toss if you chose neither, would you feel that it would be just as good to leave it to a coin toss as to pick the first list based on the paucity of sequences of numbers in a row and the paucity of round (10,20,30) numbers?

Here's the code--it should run in MatLab (Octave is basically a freeware version of MatLab)--pointing out mistakes to me would be appreciated:

function dummy = getdist(n)
    tic;
    for m=1:n
        a=zeros(21,5);
        for i=1:21
            a(i,1) = fix(1+rand*35);
            for j = 2:5
                rn = fix(1+rand*35);
                while any(rn==a(i,1:j-1))
                    rn = fix(1+rand*35);
                end
                a(i,j)=rn;
            end
        end
        a=sort(a,2);
        nrounds(m)=sum(10==a(:)) + sum(20==a(:)) + sum(30==a(:));
        nseq2(m)=0;
        for i=1:21
            b=diff(a(i,:));
            nseq2(m)=nseq2(m)+sum(1==b);
        end
        
    end


    for i=0:6
        totalrounds(i+1)=sum(i==nrounds);
    end
    totalrounds=totalrounds
    
    for i=0:6
        totalseq2(i+1)=sum(i==nseq2);
    end
    totalseq2=totalseq2
    
    for i=0:6
        total(i+1)= sum( nseq2<=i & nrounds<=i );
    end
    total=total
    
    toc
end