  We introduce a stochastic model of binary opinion dynamics in one dimension.
The binary opinions $\pm 1$ are analogous to up and down Ising spins and in the
equivalent spin system, only the spins at the domain boundary can flip. The
probability that a spin at the boundary is up is taken as $P_{up} = \frac
{s_{up}} {s_{up} + \delta s_{down}}$ where $s_{up} (s_{down})$ denotes the size
of the domain with up (down) spins neighbouring it. With $x$ fraction of up
spins initially, a phase transition is observed in terms of the exit
probability and the phase boundary is obtained in the $\delta -x$ plane. In
addition, we investigate the coarsening behaviour starting from a completely
random state; conventional scaling is observed only at the phase transition
point $\delta = 1$. The scaling behaviour is compared to other dynamical
phenomena; the model apparently belongs to a new dynamical universility class
as far as persistence is concerned although the dynamical exponent, equal to
one, is identical to a similar model with no stochasticity.
