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Insoluble marbles-in-urn problem?

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  • Subject: [mg119982] Insoluble marbles-in-urn problem?
  • From: John Feth <johnfeth at>
  • Date: Sun, 3 Jul 2011 04:11:34 -0400 (EDT)

There is a huge urn full of marbles, each marked with a single digit:
0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.  The marked marble quantities are
uniformly distributed between all of the digits and the marbles are
thoroughly mixed.  You look away, choose 10 marbles, and put them in a
black velvet bag.

When you have some time, you look away, open the bag, and remove one
marble.  You close the bag, look at the digit on the marble, open a
beer perhaps, and calculate the probability that there is at least one
more marble in the bag with the same digit.

The answer is brute forced below is there a formal way to obtain the
answer?  I don't believe the marbles-in-urn standby, the
hypergeometric distribution, is any help at all.

Copy and paste the algorithm below into Mathematica (V6 or newer) to
find the surprising answer, estimated from a million tests in about 16


Below are some definitions that might make the algorithm above a
little less opaque.

rag is a table of 10 digit random strings below


bug is a table of which digit to pick from each rag[ [ ] ] above, i.e.
the 4th from the left in rag[[1]], the 2nd from the left in rag[[2]],


selection is a table of the values of the digit picked above, i.e.,
the 4th digit in rag[[1]] is a 3, the 2nd digit in rag[[2]] is a 5,


freq is a table of the number selected digits in rag[[n]], i.e., there
are two 3s in rag[[1]], three 5s in rag[[2]], one 9 in rag[[3]], etc.


bull tallies how many times the chosen digit occurs


bullsort tallies the number of times the chosen digit occurs; the
chosen digit occurred once one time (9's above), twice one time (4's
above), and once three times (5's above)


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