Which Loom Wove This Cloth?

A mill runs two looms. Flaws appear along the cloth as a Poisson process:

• Loom A produces flaws at rate $\lambda_A = 1$ per metre.
• Loom B produces flaws at rate $\lambda_B = 3$ per metre.

Loom A weaves 70% of all cloth, loom B the remaining 30%. A one-metre piece is picked at random from the mill's output, and it is found to have exactly 2 flaws.

Recall the Poisson probability mass function:

$P(X = k) = \frac{e^{-\lambda}\lambda^{k}}{k!}$

Useful values: $e^{-1} \approx 0.3679$, $\quad e^{-3} \approx 0.0498$.

What is the posterior probability that the piece came from loom A?