2019 NJC P2 Q5

The random variable \(X\) is the number of successes in \(n\) independent trials of an experiment in which the probability of a success at any one trial is \(p\).

1. Show that \(\frac{\mathrm{P}\left( X=k+1 \right)}{\mathrm{P}\left( X=k \right)}=\frac{\left( n-k \right)p}{\left( k+1 \right)\left( 1-p \right)}\), where \(k=0,1,2,...,\left( n-1 \right)\).[2]
2. Using the result in part (i), find the most probable number of successes when \(n=99\) and \(p=\frac{7}{9}\).[3]