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Question

A bag contains a certain number of balls, some of which are white; a ball is drawn and replaced, another is then drawn and replaced; and so on: if is the chance of drawing a white ball in a single trail, find the number of white balls that is most likely to have been drawn in n trails.

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Solution

The chance of drawing exactly r white balls is nCrprqnr, and we have to find for what value of r this expression is greatest.
Now nCrprqnr>nCr1pr1qn(r1),
so long as (nr+1)p>rq,
or (n+1)p>(p+q)r.
But p+q=1; hence the required value of r is the greatest integer in p(n+1).
If n is such that pn is an integer, the most likely case is that of pn successes and qn failures.

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