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.

Answer 1

The chance of drawing exactly r white balls is ${}^n{C}_r{p}^r{q}^{n-r},$ and we have to find for what value of $r$ this expression is greatest.
Now ${}^n{C}_r{p}^r{q}^{n-r}>{}^n{C}_{r-1}{p}^{r-1}{q}^{n-(r-1)},$
so long as $(n-r+1)p>rq,$
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.