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N dice (Posted on 2017-05-25) Difficulty: 3 of 5
In "5 dice" Andy had five regular dice. Now he has a total of N regular dice. He claims that the odds of rolling exactly M sixes is exactly half as likely as rolling (M-1) sixes. (M < N).

For what values of N is this true?

State the pattern if there is one.

Express M as a function of N.

No Solution Yet Submitted by Larry    
Rating: 3.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer aided solution Comment 3 of 3 |
N=7k-1 where k is a positive integer. And M=2k using the same k, so M=2(N+1)/7.

The program output shows N, M-1, M, p(M-1), p(M) and the ratio of the two probabilities. 

6  1  2  0.401877572016  0.200938786009  1.99999999999
13  3  4  0.213845355123  0.106922677562  1.999999999991
20  5  6  0.12941029198  0.06470514599  2
27  7  8  0.082745419868  0.041372709934  2
34  9  10  0.054558633927  0.027279316963  2.000000000037
41  11  12  0.036686982814  0.018343491407  2
48  13  14  0.025008402857  0.012504201428  2.00000000008

from

p1=1/6
p2=5/6
Open 2,"n dice.txt","w"
for n=3 to 50
   for x=1 to n
     p=prob
     prob=comb(n,x)*p1^x*p2^(n-x)
     if prob<>0 then
     if abs(p/prob-2) <.00000001 then
        Print n,x-1,x,p,prob,p/prob
        o$=str$(n)+"  "+str$(x-1)+"  "+str$(x)+"  "+str$(p)+"  "+str$(prob)+"  "+str$(p/prob)
        writeln 2,o$
     Endif
     endif
   next
next

Close 2
end

sub fact(x)
  f=1
  For i=2 to x
    f=f*i
  Next
return(f)
end sub

sub comb(n,r)
  c=fact(n)/(fact(r)*fact(n-r))
  Return c
End Sub

  Posted by Charlie on 2017-05-25 12:14:14
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