[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 proved a theorem often called the “Weak Law of

Large Numbers.” Most people's intuition and our computer simulations suggest that, if we toss a coin a sequence of times, the proportion of heads will really approach 1/2; that is, if [math]S_n[/math] is the number of heads in [math]n[/math] times, then we will have

[[math]] A_n = \frac {S_n}n \to \frac 12 [[/math]]

as [math]n \to \infty[/math]. Of course, we cannot be sure of this since we are not able to toss the coin an infinite number of times, and, if we could, the coin could come up heads every time. However, the “Strong Law of Large Numbers,” proved in more advanced courses, states that

[[math]] P\left( \frac {S_n}n \to \frac 12 \right) = 1\ . [[/math]]

Describe a sample space [math]\Omega[/math] that would make it possible for us to talk about the event

[[math]] E = \left\{\, \omega : \frac {S_n}n \to \frac 12\, \right\}\ . [[/math]]

Could we assign the equiprobable measure to this space? \choice{}{(See Example.)}