[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A drug is assumed to be effective with an unknown probability

[math]p[/math]. To estimate [math]p[/math] the drug is given to [math]n[/math] patients. It is found to be effective for [math]m[/math] patients. The method of maximum likelihood for estimating [math]p[/math] states that we should choose the value for [math]p[/math] that gives the highest probability of getting what we got on the experiment. Assuming that the experiment can be considered as a Bernoulli trials process with probability [math]p[/math] for success, show that the maximum likelihood estimate for [math]p[/math] is the proportion [math]m/n[/math] of successes.