  Many types of bacteria can survive under stress by switching stochastically
between two different phenotypes: the "normals" who multiply fast, but are
vulnerable to stress, and the "persisters" who hardly multiply, but are
resilient to stress. Previous theoretical studies of such bacterial populations
have focused on the \emph{fitness}: the asymptotic rate of unbounded growth of
the population. Yet for an isolated population of established (and not very
large) size, a more relevant measure may be the population \emph{extinction
risk} due to the interplay of adverse extrinsic variations and intrinsic noise
of birth, death and switching processes. Applying a WKB approximation to the
pertinent master equation of such a two-population system, we quantify the
extinction risk, and find the most likely path to extinction under both
favorable and adverse conditions. Analytical results are obtained both in the
biologically relevant regime when the switching is rare compared with the birth
and death processes, and in the opposite regime of frequent switching. We show
that rare switches are most beneficial in reducing the extinction risk.
