Choosing randomly two integer numbers from 10 to 1,000,000 inclusive,
What are the chances they will be represented:
a. each by two distinct digits, albeit different
for each one of the numbers e.g. 322 and 78778
b.exactly the same two, like 3334 and 433344
or
c. distinct couples, sharing one digit like 5558 and 88338
There are 999,991 numbers in the range, and the square of this is the denominator of each pair's probability, as the same number could be in both choices.
clearvars,clc
pairCt=zeros(10,10);
for n=10:1000000
ns=unique(char(string(n)));
if length(ns)==2
s1=str2double(ns(1))+1;
s2=str2double(ns(2))+1;
pairCt(s1,s2)=pairCt(s1,s2)+1;
end
end
pairCt
builds and displays the number (ways) of each such pair combinations:
low \ high digit
digit \ 1 2 3 4 5 6 7 8 9
0 0 58 57 57 57 57 57 57 57 57
1 0 0 114 114 114 114 114 114 114 114
2 0 0 0 114 114 114 114 114 114 114
3 0 0 0 0 114 114 114 114 114 114
4 0 0 0 0 0 114 114 114 114 114
5 0 0 0 0 0 0 114 114 114 114
6 0 0 0 0 0 0 0 114 114 114
7 0 0 0 0 0 0 0 0 114 114
8 0 0 0 0 0 0 0 0 0 114
9 0 0 0 0 0 0 0 0 0 0
Then this table is utilized by
allNos=1000000-10+1;
parta=0; partb=0; partc=0;
for a=0:8
s1=a+1;
for b=a+1:9
s2=b+1;
partb=partb+pairCt(s1,s2)^2;
for x=0:8
s3=x+1;
for y=x+1:9
s4=y+1;
u=unique([a b x y]);
if length(u)==4
parta=parta+pairCt(s1,s2)*pairCt(s3,s4);
end
if length(u)==3
partc=partc+pairCt(s1,s2)*pairCt(s3,s4);
end
end
end
end
end
parta=sym(parta)/allNos^2
partb=sym(partb)/allNos^2
partc=sym(partc)/allNos^2
parta=eval(parta)
partb=eval(partb)
partc=eval(partc)
parta=1/(parta)
partb=1/(partb)
partc=1/(partc)
which multiplies the ways of the first chosen number by the ways of the second.
Numbers are chosen in both directions and so do not need to be doubled; when they are the same, of course, they are counted once.
The results are:
Exact probabilities:
parta =
13106352/999982000081
partb =
497212/999982000081
partc =
7722360/999982000081
Decimal:
parta =
1.31065879175209e-05
partb =
4.97220949936824e-07
partc =
7.72249900435656e-06
Reciprocal (i.e., 1 in ...)
parta =
76297.5082678231
partb =
2011178.33053305
partc =
129491.761596325
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Posted by Charlie
on 2023-07-14 11:56:35 |
Solution
