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Question
If mn coins have been distributed into na purses, n into each, find (1) the chance that two specified coins will be found in the same purse; and (2) what the chance becomes when r purses have been examined and found not to contain either of the specified coins.
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Solution
Represent the two coins by A and B. then the chance that B is with A is n−1mn−1 for whenever A is placed, there remain mn−1 possible position for B, n−1 of which are favourable.
Hence the chance that A and B are not together is n(m−1)mn−1Now consider the m−r purses which have not been examined.
If A and B are together, the chance that they occur in these purses is m−rm.
If A and B are apart the chance that they occur in these purses is (m−r)(m−r−1)m(m−1);
For m(m−1) is the total no. of ways in which they can occur separately in any 2 purses whatever and (m−r)(m−r−1) is the no. of ways in which they can occur separately in any two of the purses we are considering
Hence the required chance,=n−1mn−1.m−rm÷{n−1mn−1m−rm+n(m−1)mn−1(m−r)(m−r−1)m(m−1)}=n−1mn−nr−1
Represent the two coins by A and B. then the chance that B is with A is n−1mn−1 for whenever A is placed, there remain mn−1 possible position for B, n−1 of which are favourable.
Hence the chance that A and B are not together is n(m−1)mn−1
Now consider the m−r purses which have not been examined.
If A and B are together, the chance that they occur in these purses is m−rm.
If A and B are apart the chance that they occur in these purses is (m−r)(m−r−1)m(m−1);
For m(m−1) is the total no. of ways in which they can occur separately in any 2 purses whatever and (m−r)(m−r−1) is the no. of ways in which they can occur separately in any two of the purses we are considering
Hence the required chance,
=n−1mn−1.m−rm÷{n−1mn−1m−rm+n(m−1)mn−1(m−r)(m−r−1)m(m−1)}
=n−1mn−nr−1
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