=== Model: grok-4.6-direct | Task: bench:gpqa_diamond | Rep: 1 ===
=== Score: 5/5 | Wall: 14.769s | Cost: $0.008048 ===
=== Pass: True | Finish: stop | In-reasoning-only: False ===

--- JUDGE ---
gpqa_diamond

--- RESPONSE ---
The geometric transit probability for a circular orbit is \(P_\mathrm{transit} \approx R_\star/a\). The two host stars have identical radii, so the ratio of transit probabilities simplifies to \(P_1/P_2=a_2/a_1\).

Kepler’s third law gives \(a\propto(P^2M_\star)^{1/3}\). Substituting the given period ratio \(P_1=P_2/3\) and mass ratio \(M_{\star1}=2M_{\star2}\) yields
\[
\frac{a_2}{a_1}=\Bigl(\frac{9}{2}\Bigr)^{1/3}\approx1.65.
\]
Hence Planet 1 is \(\approx1.65\) times more likely to transit.

The answer is (B)