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Balls2
*To*: mathgroup at smc.vnet.net
*Subject*: [mg31366] Balls2
*From*: "Juan Erfá" <erfa11 at hotmail.com>
*Date*: Wed, 31 Oct 2001 03:30:56 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Hello.
In my letter from yesterday there is a fail.
The function F2 shoud be:
F2[b_,x_]:=Sort[F[b,x],Greater], and not Sort[F1[b,x],Greater]
The function F, is define dawn in the page and it have to be in the
begining.
The all letter shoud be like that:
<< DiscreteMath`Combinatorica`
The problen is how to share b balls in x boxes.
There is 4 cases:
In[1]:= F[b_, x_] := (q = Table[0, {x}];
Table[r = Random[Integer, {1, x}]; q[[r]]++, {b}]; q)
1.The balls are equals, the boxes are diferent
In[2]:= F1[b_,x_]:=RandomComposition[b,x]
2.The balls are equals, the boxes are equals
In[3]:= F2[b_,x_]:=Sort[F[b,x],Greater]
3.The balls are diferents, the boxes are equals
In[4]:= F3[b_, x_] := (a3 = F2[b, x]; r3 = RandomTableau[a3];
PadRight[r3, x, {{}}])
4.The balls are diferents, the boxes are diferents
In[5]:= F4[b_, x_] := (a4 = F3[b, x]; r4 = RandomPermutation[x];
Map[a4[[#]] &, r4])
In[6]:= F1[11,7]
Out[6]= {0,2,0,3,5,1,0}
In[7]:= F2[11,7]
Out[7]= {5,2,2,1,1,0,0}
In[8]:= F3[11,7]
Out[8]= {{1,4,6,11},{2,5},{3,8},{7,9},{10},{},{}}
In[9]:= F4[11,7]
Out[9]= {{2,6},{3},{10},{},{1,4,5,7,9,11},{},{8}}
I have also calculate the probability to share b balls in x boxes, and n
balls at least in one box.
In[12]:= Prob[b_, x_, n_] := (t = Table[F[b, x], {200}];
c = Count[Map[MemberQ[#, n] &, t], True]; 0.005*c)
The birthday problem is:
In[13]:= Prob[23,365,2]
Out[18]= 0.515
I get this results:
In[14]:= Prob[90,365,3]
Out[14]= 0.515
In[15]:= Prob[190,365,4]
Out[20]= 0.505
The function BN[x_,n_,p_]= b,
(to get b, knowing x, n, and the probability p)
If n=2, then p=1-x!/((x-b)! x^b), and we can solve this ecuation for b
In[23]:= B2[x_,p_]:=FindRoot[1-x!/((x-b)!x^b)==p,{b,1,x}]
In[24]:= B2[365,.5]
Out[24]= {b->22.7677}
If n>2, then I don't know.
Salutes.Juan
PD.I don't know how to pas the programs I write in Mathematica to the
e-mail, so I have to copy and paste every sentens, an after to correct the
signs that can not be convert, and I get confuse.
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