Cracks Worth Chasing

A pipeline company trains a model to spot cracked welds in X-ray photos. In a test set of 600 photos, 60 welds are truly cracked. Every photo the model flags gets a second look from an inspector, so a false alarm costs a few minutes of their time. A missed crack can mean a leak.

The model outputs a crack probability, and two thresholds are on the table:

F1 treats a missed crack and a false alarm as equally bad. The team wants a score that cares more about Recall, so it uses the F-beta score:

$$F_\beta = \frac{(1 + \beta^2)\cdot \text{Precision} \cdot \text{Recall}}{\beta^2 \cdot \text{Precision} + \text{Recall}}$$

With $\beta = 1$ this is exactly F1. A larger $\beta$ gives Recall more say. The team uses $\beta = 2$.

Which threshold has the higher $F_2$, and what is that $F_2$? Enter the $F_2$ as a decimal between 0 and 1, rounded to 3 decimal places.