  For ergodic systems with generating partitions, the well known result of
Ornstein and Weiss shows that the exponential growth rate of the recurrence
time is almost surely equal to the metric entropy. Here we look at the
exponential growth rate of entrance times, and show that it equals the entropy,
where the convergence is in probability in the product measure. This is however
under the assumptions that the limiting entrance times distribution exists
almost surely. This condition looks natural in the light of an example by
Shields in which the limsup in the exponential growth rate is infinite almost
everywhere but where the limiting entrance times do not exist. We then also
consider $\phi$-mixing systems and prove a result connecting the R\'enyi
entropy to sums over the entrance times orbit segments.
