A coincidence becomes evidence through a specified search

For the event on 17 August 2017, the LIGO–Virgo, Fermi Gamma-ray Burst Monitor and INTEGRAL joint paper published in October 2017 reported GRB 170817A arriving 1.74±0.05 seconds after GW170817. This is a dated result from that primary analysis, not a universal merger-to-gamma emission interval. The joint directional and temporal evidence supports a common event and shows that binary neutron-star mergers can produce short gamma-ray bursts. It does not establish that every short burst has an identical central engine, viewing angle or intrinsic delay.

An arrival delay can contain intrinsic source emission delay, geometry within the source and a difference in propagation times. Cosmological time stretching multiplies a host-frame intrinsic interval by one plus redshift. A bound on different messenger speeds therefore requires assumptions or constraints on the intrinsic contribution and the propagation model. Multiplying an arrival delay by c does not uniquely measure an engine radius. For a relativistic outflow, differing emission locations and photon paths can contribute even if all relevant signals propagate at their standard speeds.

Suppose a prespecified time-and-direction window has expected background count λ. Under a Poisson background model, the chance of at least one event is one minus exp(−λ). This is a probability of data under a background model, rather than the probability that the background hypothesis is true. Selecting a window after inspecting an event, scanning many sources or trying many energy cuts changes the relevant search space. A reported local significance cannot be carried into a population claim without accounting for those trials.

For M independent equal background searches, the probability of one or more windows producing an event is one minus exp(−Mλ). Real windows can overlap and share candidates, making the independence assumption false; calibrated simulations or time shifts can then estimate the full search statistic. Detector duty cycles, localization errors and event-quality selection also enter. A low background rate supports association only relative to the signal model and its prior plausibility. Neither a p-value nor a likelihood ratio alone supplies a complete posterior source probability.

A joint likelihood can combine independent detector data conditional on shared source parameters, while marginalizing uncertain distance, inclination, emission parameters and calibration. Conditional independence is a modeling claim, not automatic when instruments share backgrounds or selection information. For neutrinos, the expected count is the energy integral of number fluence times effective area, summed over flavors and evaluated for the source direction. A null result constrains that response-weighted fluence under an assumed spectrum and duration; it does not directly rule out all baryonic acceleration or fix bolometric source power.

The probability lab illustrates repeated opportunities for a false flag. Reset and set both controls to p=0.01,n=1, predicting zero flags with probability 0.99. Reset and set p=0.01,n=30, predicting zero flags with probability 0.99³⁰≈0.740, hence at least one with probability about 0.260. These are exact independent Bernoulli trials, not the Poisson windows or the actual multimessenger search. The lesson is to specify the trial family before interpreting an apparent coincidence; the instrument has no sky map, detector selection or astrophysical prior.