  We show that the persistence probability $P(t,L)$, in a coarsening system of
linear size $L$ at a time $t$, has the finite size scaling form $P(t,L)\sim
L^{-z\theta}f(\frac{t}{L^{z}})$ where $\theta$ is the persistence exponent and
$z$ is the coarsening exponent. The scaling function $f(x)\sim x^{-\theta}$ for
$x \ll 1$ and is constant for large $x$. The scaling form implies a fractal
distribution of persistent sites with power-law spatial correlations. We study
the scaling numerically for Glauber-Ising model at dimension $d = 1$ to 4 and
extend the study to the diffusion problem. Our finite size scaling ansatz is
satisfied in all these cases providing a good estimate of the exponent
$\theta$.
