Posterior probability of a factory given a long bulb lifetime

Bulbs produced by a factory $F_i$ have lifetimes (in months) distributed as $\mathrm{Exp}\left(\frac1{3^i}\right)$ for $i=1,2,3$. A firm randomly procures $40\%$ of its required bulbs from $F_1$, $30\%$ from $F_2$ and $30\%$ from $F_3$. A randomly selected bulb from the firm is found to be working after $27$ months. The probability that it was produced by the factory $F_3$ is ____________