Reply to “Comment on ‘The Reduced Detection Rate of Signals That Are Hidden by Earthquakes: Case Studies with Spotlight Detectors That Operate at Seismic Arrays,’ by Joshua D. Carmichael, Brent G. Delbridge, and Richard Alfaro-Diaz” by Paul G. Richards

ABSTRACT We reply to a comment that Paul G. Richards (Richards, 2026) directed to an article by Carmichael et al. (2025). Richards expressed concern over two issues. The first issue relates to the size of the explosions discussed in the article. The second relates to the probability that earthquakes and explosions sourced near the same location coincide in time, by chance. We do not address the first issue because the original article does not assign significance to the size of the explosions from the point of view of an experimental party. We do respond to Richard’s comment on the second issue and claim that it can be explained through a clarification about conditional versus joint probabilities. Our present discussion, therefore, describes the difference between the conditional probability that a correlator fails, given that an explosion and earthquake coincidently occur in the same place, and the joint probability that a correlator fails while an earthquake and explosion also coincidentally occur in the same place. This latter, joint probability appears to be rare. We clarify that our original article did not treat the joint probability, only the conditional probability.