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 p be the chance of white balls that is most likely to have been drawn in n trials. For $p=\frac{1}{2}$ and $n=12$, the number of white balls required to be drawn is $k$. Find the value of $k$.

Answer 1

The probability of drawing exactly r white balls in n trials is
${}^n{C}_r{q}^{n-r}$ ,
and it is required to find for what value of r this expression is greatest.
Now ${}^n{C}_r{p}^r{q}^{n-r}{>}^n{C}_{r-1}{p}^{r-1}{q}^{(r-1)}$
So long as $\frac{n!}{r!(n-r)!}p>\frac{n!}{(r-1)!(n-r+1)!}q$
Or $(n-r+1)p>rq$
or $(n+1)p>(p+q)r.$
or $np+p>r[\because p+q=1]$
Hence $r>7/2$
Therefore $k=4$