Trying to Calculate Basic Lunar Probabilities I'm trying to do what I thought was a basic problem but doesn't seem to be working out properly.  I'm looking at disproving a claim that tides cause earthquakes, where the main mechanism that people claim is that it happens during a new or full moon, and especially during a perigee moon (when it's closest to Earth).
People like to give ±1 week windows which is basically half a lunar cycle, so it seems basic that the probability an earthquake would happen, given random chance, would be 50% that it would happen within a week of, e.g., a full moon.  Similarly, since perigee/apogee happens roughly on the same timescale, it seems as though it would be 50% chance that an earthquake would happen within 1 week of a perigee moon.  Put the two together and you have a 25% chance that an earthquake would happen within both a perigee and full moon, correct?
(Important note, updated:  Actual period between new/full moons averages 29.52 days, actual time between perigee moons averages 27.56 days.  However, apogee and perigee are NOT normally distributed.  Apogee happens a mode of 27.78 days (mean is 27.55±0.27), while perigee is much more asymmmetric, having a mode of 28.4 days at the peak of approximately a Lorentzian.  Half the max of the Lorentzian is ±0.08 days.  But, the mean is 27.56 with a standard deviation of 1.12; the range is 24.6-28.6 days.  I'm thinking this could throw off the modeling?)
Assuming that's correct, I seem to be running into issues when trying to figure out the probability that an earthquake would happen by chance within ±X days of BOTH a perigee and full moon.  I thought the equation would be simply (2*X/(# days in lunar month, 29.5))*(2*X/(# days between perigee, 27.5)).
However, when I do a monte carlo simulation with 500,000 randomly chosen dates within the time period of 1933 to 2012 (just happens to be when I have earthquake data), the fractions do not line up.  For example, the simulation shows that 14.4% should be within 5 days of both a full and perigee moon, but my above math says it's only 12.3%.
I have checked the results of my simulation against days from perigee, apogee, new moon, and full moon times.  As expected from a random distribution with a large $N$, the number of times that the simulated earthquake is a given time period away from maximum perigee/apogee/new/full moon is even.  Except for perigee, where I see a fall-off for $>|±12.5|$ days from when it's closest to perigee.  I'm thinking this has to do with the non-Gaussian distribution of perigee times?  And could that account for the 2% difference at the 5-day example?
Is the best way to approach this, because these perigee times are a bit crazy, to simply go with the Monte Carlo results?
P.S. This has been updated to better reflect a correction I made in my data.  When I initially posted this, I had some incorrect full/new moon dates in my table that were throwing some results off.
 A: There is an oddity that $29.52/27.56$ is unusually close to $15/14.$ So, even though the proportion of times within $5$ days of both will eventually be about $\frac{2X}{29.52} \frac {2X}{27.56}$, it might take longer to converge than you expect. Instead of trying to calculate by a formula, you can simply ask a computer to count how many minutes are within $X$ of the perigee and full moon over that whole range. However, I don't think this effect is large enough to explain the difference between $15.2\%$ and $12.3\%$.
A Monte Carlo simulation normally would mean that you choose times randomly, not evenly spaced. One possibility is that you chose a gap which is too close to a simple rational times one of the periods. $79$ years/$43000$ is $1.49$ days, which might be too close to $29.52/20 = 1.48$. If you used exactly $29.52/20$ then instead of $10/29.52 = 33.88\%$ samples within $5$ days you would get $7/20 = 35\%.$ Again, this doesn't seem to be large enough to explain the discrepancy, but why add significant errors of this type?
