# Test for independence, when I'm missing one bin of a 2x2 contingency

I'm looking for some help devising a hypothesis test for the following situation.

1. I have a radioactive source that spits out a particle every so often.

2. Also, I have two particle detectors: a red particle detector, and a green particle detector. Whenever the red particle detector detects a particle, it flashes a red light; let $R$ denote the event that the particle was detected by the red detector, and $r$ the complement event that the particle was not detected by the red detector. Whenever the green particle detector detects a particle, it flashes a green light; let $G$ be the event that the green detector detects the particle, and $g$ that it does not. Thus, each emitted particle falls into one of four categories:

• detected by both detectors ($RG$),
• detected by the red detector but not the green detector ($Rg$),
• detected by the green detector but not the red detector ($rG$), or
• not detected by either detector ($rg$).
3. Each time a particle is emitted, the red detector has some probability of detecting the particle, and the green detector has some probability of detecting the particle. (They will never trigger a false detection when no particle is present.) I know that each particle is handled identically and independently of all other particles, but I don't know whether the two detectors are independent of each other. It's possible that they are independent (i.e., $\Pr[RG] = \Pr[R] \Pr[G]$), or that they are correlated (i.e., $\Pr[RG] \ne \Pr[R] \Pr[G]$); I don't know which is the case, a priori.

4. I keep a count of the number of $RG$-detections (i.e., number of times when both detectors detected something), number of $Rg$-detections (i.e., number of times when the red detector detected something, but not the green one), and the number of $rG$-detections. Unfortunately, I have no way to measure the number of $rg$-situations, since those particles aren't detected by either of the detector. At the end of the experiment, I have three non-negative integers, representing these counts.

I want to test the hypothesis $H$ that the two detectors are independent, i.e., that event $R$ is independent of event $G$. Can anyone help suggest a way to compute the $p$-value of this hypothesis, given 3 numbers from such an experiment?

I'd be perfectly satisfied with a computer algorithm/procedure to compute the $p$-value. I don't need a simple formula; something that could be computed by a computer would be sufficient.

Here's another way to view this. We could form a 2x2 contingency table, such as this one:

     G |  g
---------
R | 17   22
r | 12    ?


recording that we saw 17 $RG$-events, 22 $Rg$-events, and so on. Unfortunately, the lower-right cell is empty, since we don't know how many $rg$-particles were emitted. If we had counts for all four cells, presumably we could use Fisher's exact test, but we don't. Also, we aren't given $\Pr[R]$ or $\Pr[G]$ (I guess they are nuisance parameters) or the total number of particles emitted.

Any suggestions?