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Only two digits (Posted on 2023-07-14) Difficulty: 3 of 5
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

No Solution Yet Submitted by Ady TZIDON    
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Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts simulation | Comment 2 of 5 |
This program has a bug, highlighted below, and corrected in my later post. Therefore the results are wrong.


clearvars,clc
typea=0; typeb=0; typec=0;
for trial=1:100000000
  n1=char(string(randi(999991)+9));
  if length(unique(n1))==2
    n2=char(string(randi(999991)+9));
    if length(unique(n1))==2
      switch length(unique([n1 n2]))
        case 2
          typeb=typeb+1;
        case 3
          typec=typec+1; 
        case 4
          typea=typea+1;
      end
    end
  end
end
  typea
  typeb
  typec
  typea/trial
  typeb/trial
  typec/trial
  trial/typea
  trial/typeb
  trial/typec          

provides a limited view of the probabilities:

In 100 million trials, these are the numbers of cases of the three types:

typea =
       40367
typeb =
    62
typec =
        3266

That's enough to get about 2-digit accuracy in types a and c, and less than that in type b.

Those probabilities were calculated at:

ans =
                0.00040367
ans =
                   6.2e-07
ans =
                 3.266e-05

and the  reciprocals (for 1 in ... form):


ans =
          2477.27103822429
ans =
          1612903.22580645
ans =
          30618.4935701163

Both sets are a, b, c order.

Edited on July 14, 2023, 11:49 am
  Posted by Charlie on 2023-07-14 09:37:54

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