N1982 P1 Q13

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

Show that

\(\frac{\mathrm{P}\left( X=k+1 \right)}{\mathrm{P}\left( X=k \right)}=\frac{\left( n-k \right)p}{\left( k+1 \right)\left( 1-p \right)},k=0,1,2,...,\left( n-1 \right)\)

Find the most probable number of success when \(n=10\) and \(p=\frac{1}{4}\).