A reader writes:
Re "If it looks random, it probably isn't"; can I presume that I can apply the same logic instead of lightning to missiles ?
Meaning if a missile falls in a location is it highly probably that another will fall in the same area?
Right at the top, I'm just going to say that I'm not going to say anything about the politics of this situation which has been particularly in the news, yet again, in the last few days, and I'd appreciate it if commenters didn't either. I sure do have opinions on this subject, but there are a million places people can have arguments about the Heroic Downtrodden Palestinians versus the Stoic Peace-Loving Israelis, and this blog post is not the time nor the place.
(Readers who feel an uncontrollable need to argue about something are encouraged to do so on that post about a free book that some guy argued against without even looking at the free book, that post about the existence or otherwise of "copper bullion" where a lady turned up to hotly argue that buying copper by the ounce is a great idea, and the few posts that sort of ended up being about Jock Doubleday and the floridly preposterous conditions of his "vaccine challenge". Bonus points for anybody who manages to persuade me to their religion, or that climate change isn't happening, or that every man secretly craves sex with other men.)
Speaking of time and place, though, both things are important here, because impacts of artillery over time have a distribution both in space and in time. This is a problem which has been addressed before, most famously in analyses of where the thousands of V-1 and V-2 missiles landed in and around London in World War II.
(That, by the way, is one of the approximately 300 themes of Thomas Pynchon's Gravity's Rainbow, which I read once, and finished, purely because I am susceptible to the sunk-costs fallacy. Forget Israelis and Palestinians - if you want to see true hatred, ask me what I think of Gravity's Fucking Rainbow. How can such dazzling ability to paint a whole scene with ten words be used to create such an indigestible housebrick of a book?!)
The British analysts in WWII wanted to know how good the guidance systems of the V-2, in particular, were. Since it turned out that V-2 hits conformed quite well to a random Poisson distribution, the analysts reached the correct conclusion that the V-2 did not have a guidance system capable of targeting particular areas of the city.
Even the V-1s did have some kind of guidance system, though. There's a popular belief that V-1s "flew until their fuel ran out", but they actually had an autopilot that counted the revolutions of a little propeller on the nose of the missile, and put the bomb in a dive after a predetermined flight distance. That dive happened to cut off the fuel flow and stop the engine in early V-1s, creating the out-of-fuel legend; this bug was fixed in later V-1s, most of which therefore managed to do their final dive under power, as originally designed.
Aaaaanyway, if you abstract it all out and presume that any arbitrary square metre of Sderot is as likely to be hit by a rocket from the Gaza Strip as any other over a given hour, then the lightning-strike conditional-probability situation applies. Whichever square metre you're standing in is, by these assumptions, as likely to be hit over the next hour as any other, but in order for a missile to next hit your particular square metre two hours from now, there must by definition not be a hit in the next hour. So, as in the lightning-strike example, you multiply the probability of no-hit next hour by the probability of a hit the hour after that, and get a slightly lower number. The probability of a hit in any given spot in any given hour is the same, but the probability of the next hit being separated from now by one, or ten, or a million, hours gets lower and lower as time wears on.
This is of no use whatsoever in determining what location's going to be hit next, though; we assumed right at the start of our abstraction that the missiles were falling randomly. It just explains why clusters of hits, close in time and/or space, can and will occur - and encourage the statistically untutored to explain them in terms of aim and guidance systems - even if the actual distribution of events is random.
In the real world, of course, the distribution of Palestinian missile hits on Israel is only partially random. The basic garage-built missiles have no guidance system at all and variable performance characteristics, which predisposes them to land in a Poisson distribution just like the V-2s in WWII. But there are many launch sites and several other kinds of missile...
image source: Meir Amit Intelligence and Terrorism Information Center)
...making the overall situation extremely complex.
Many Palestinian missiles are just shot in the right general direction (in the opinion of the people launching them, of course). In videos of the Iron Dome missile-defence system in operation...
...you'll occasionally see the system not bothering to shoot an interceptor missile at some of the incoming fire, because the system calculates that that rocket isn't going to hit a populated area. (See also, mortar attacks by insurgents/terrorists/freedom-fighters, strike out whichever does not in your opinion apply, on military bases and police stations and various other targets in Northern Ireland, Afghanistan, Iraq, and a dismayingly long list of other places.)
But the more sophisticated actual military rockets definitely can be aimed, at least to some extent, so the distribution of their hits will be skewed toward populated areas, military targets, or whatever else the people launching them are trying to hit. You could probably protect yourself from those rockets by a considerable amount by going and living in a tent in the middle of the desert.
As regards avoiding getting blown up in the Israeli towns within rocket range of the Gaza Strip, though, I'm afraid conditional probability has nothing to offer you. All it tells you is that it's improbable that some particular location will not be hit for an extended period of time, which you already, unfortunately, knew.