Worked Example 5

In a binomial probability distribution there are $n$ trials and the probability of success for each trial is $p$. If the mean is $8$ and the variance is $4.8$, find the value of $n$ and $p$. Denoting $\text{P}\left( X=r \right)$ by ${{p}_{r}}$, show that $\frac{{{p}_{r+1}}}{{{p}_{r}}}=\frac{2\left( 20-r \right)}{3\left( r+1 \right)}$, where $r=0,1,2,3,…..,18,19$.

Hence find the mode of $X$.