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 \(X\).