[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]

We have two coins: one is a fair coin and the other is a coin that

produces heads with probability 3/4. One of the two coins is picked at random, and this coin is tossed [math]n[/math] times. Let [math]S_n[/math] be the number of heads that turns up in these [math]n[/math] tosses. Does the Law of Large Numbers allow us to predict the proportion of heads that will turn up in the long run? After we have observed a large number of tosses, can we tell which coin was chosen? How many tosses suffice to make us 95 percent sure?