  Consider a coin tossing experiment which consists of tossing one of two coins
at a time, according to a renewal process. The first coin is fair and the
second has probability $1/2 + \theta$, $\theta \in [-1/2,1/2]$, $\theta$
unknown but fixed, of head. The biased coin is tossed at the renewal times of
the process, and the fair one at all the other times. The main question about
this experiment is whether or not it is possible to determine $\theta$ almost
surely as the number of tosses increases, given only the probabilities of the
renewal process and the observed sequence of heads and tails. We will construct
a confidence interval for $\theta$ and determine conditions on the process for
its almost sure convergence. It will be shown that recurrence is in fact a
necessary condition for the almost sure convergence of the interval, although
the convergence still holds if the process is null recurrent but the expected
number of renewals up to and including time $N$ is $O(N^{1/2+\alpha}), 0 \leq
\alpha < 1/2$. It solves an open problem presented by Harris and Keane (1997).
We also generalize this experiment for random variables on $L^{2}$ which are
sampled according to a renewal process from either one of two distributions.
