If something happens once every 100 years on average (i.e. a 100-year storm), what is the probability of it happening in the next 20 years?
Probability-literate Dopers, please check my reasoning: This sounds like a job for the Poisson distribution. The average number of occurences in any 20-year period is 1/5 (or 0.2), so (looking it up on a table or using the Poisson formula) the probability of exactly 1 occurence in the next 20 years is .1637, while the probability of at least one occurence is .1813.
Poisson looks about right for the problem as stated.
The one thing I’d add is that for storms, in particular, the “100-year” or “500-year” thing is a bit misleading. Really, a 100-year storm is one we estimate to have a 1 in 100 chance of happening in any given year rather than an event we expect to occur on average every century or so. That sounds the same, but really it’s a bit different.
The chances of a 1 in 100 annual event (or 1% event) happening at least once within the next 20 years can be found from a binomial distribution and is 18.2%, as contrasted by the Poisson with 18.1% for the other phrasing.
This is one of the nice things about the central limit theorem. Once we get to sufficiently large numbers of samples, the differences in how we phrase the problem end up mattering less, since they all converge to roughly the same distribution.
It matters whether you mean “Twenty years from today,” or a specific twenty year period.
NM
If the question is actually about floods and similar storm damage, rather than about figures in a vacuum: bear in mind that those percentages may have been calculated under circumstances that have changed since the calculation; or may, even if calculated recently, have been based on the previous hundred years, and circumstances may have changed over that time. Amounts and frequencies of rainfall aren’t constant, and often neither is the absorptive capacity of the soils and/or watercourses in the area.
Isn’t “twenty years from today” a specific twenty year period?
Flip it around. The probability of a 100-year storm NOT happening in any given year is 0.99.
What’s the probability of a 100-year storm NOT happening in any given 20-year period? 0.99[sup]20[/sup], or 81.8%.
So the probability of having one or more 100-year storms happening in a given 20-year period is 1- .818 = 18.2%.
All correct. However, I find that the easiest way to think about the first case (i.e. a 100-year storm) is that the probability of it happening at least once a 20 year period is exactly the same as one minus the probability of it not happening in a 20-year period, or 1 - .99[sup]20[/sup] = .182 aka 18.2%.
ETA: Machine Elf was faster than I at figuring out the vbulletin code for superscript. 
Okay, why is it not simply 20%? Why don’t we say that the chance each year is .01X20=20%?
Because those yearly chances aren’t mutually exclusive. That would be like saying "The chance for heads on each coin flip is 50%, so the chance of getting heads in two coin flips is 50% x 2 = 100%.
If it was guaranteed to happen exactly once during the 100-year period, and we just didn’t know when, then it would have a 20% chance of happening in the next 20 years.
Because that’s just not how probability works. If it were, by extension, we would also have to say that the chance over 100 years is .01x100=100%. An event that has 1/100 chance of happening every 100 years isn’t guaranteed to happen every 100 years; moreover, it especially does not have a 120% chance of happening over 120 years.
The average number of events in a 20-year period is p = 0.2. As others have implied, and assuming the events are independent, the probability of exactly k Poisson events is
… f(k) = p^k * e^-p / k!
so
f(0) = .81873075
f(1) = .16374615
f(2) = .00270671
f(3) = .00006638
and so on.
OP’s question is answered by
1-f(0) = .18126925
Not exactly the same, as noted. But it’s simple to see that they’re the same in the limit.
You considered 1-0.99[sup]20[/sup], which gives 18.209%. But we can also consider 200 stretches of 1/10th of a year, which is 1-0.999[sup]200[/sup]=18.135%. Or, 2000 stretches of 1/100th of a year, or 1-0.9999[sup]2000[/sup]=18.128%. This is very close to the Poisson result, and will get closer as the intervals get shorter.
The e[sup]-p[/sup] in septimus’ formula above is acting as a correction factor between discrete and continuous sampling. Pretty much any time you see an e in a formula, you know there’s a discrete/continuous thing going on (compound interest, etc.).
Ok, we’re basically re-stating the conclusion of the Central Limit Theorem here, i.e. whether we start with Poisson or Binomial, the distributions will end up looking pretty similar once we take enough samples.
As I noted earlier, how you pose the problem matters. If we are measuring an event that occurs, on average, once a century, Poisson is what we are measuring. But in the case of storms, it’s really a Bernoulli Random Process (i.e. Binomial) with a Poisson random variable. That is, each year is an independent trial with probably of success p but within any individual year, it’s Poisson.
But it doesn’t really matter if we are just looking for a number. Either way, we end up around 18% plus or minus a tenth of a percent, which is probably close enough for what the OP intended.
I won’t say it’s wrong to think of it that way; it’s just not the clearest approach for me personally. I think of it as illustrating the difference between discrete and continuous sampling. In the same way that a compound interest calculation converges to a certain value as the intervals decrease (annual, monthly, daily, etc.), here we have a random event that can be seen as occurring in shrinking intervals of time. The Poisson distribution is what you get as the intervals reach zero duration. And for this reason, it’s no surprise that e makes an appearance.
Not saying this is the single best way of thinking of it either; it’s just what works for me.
In the Coast Range north of San Francisco circa 1992 we suffered what were called “100-year floods” twice in three months. Similar flooding hit in 1986 (didn’t directly affect us) and 1996 (it got us). I don’t know if FEMA revised the Russian River’s 100-year floodplain criteria but I won’t be shocked if winery towns wash away within the century.
Shifting climate patterns wreck calculations based on a prior reality. We can expect an energized hydrosphere (“global warming”) to give hotter hots, colder colds, stormier storms, dryer droughts. What’s the stock prospectus caveat? “Past performance is no guarantee of future results.” That’s where 100-year floods are now.
And that’s only part of it. Many areas have built impervious surfaces (buildings, roads, parking lots, severely compacted ground) over large portions of the ground which used to absorb storm rains; and some areas have forced rivers into narrow channels, and/or destroyed barrier islands, and/or filled or even built on swampland.
I don’t think it’s been mentioned yet: The probability is also affected by whether the events are independent or not. Suppose that once an event occurs, is leaves behind some after-effect that makes the similar event either more or less likely over the next some-number of years. That changes everything, and you would have to know a bunch of the details to decide.
Consider earthquakes. Tectonic plates slide past each other, building up tension between them. Eventually, the pent-up tension gives way and a major earthquake ensues. The thinking might be that this greatly reduces the probability of another major earthquake in the same region for a number of years.
While that is unfortunate, one should always be aware that, if a one in one hundred year event happens this year, the single future year in which it most likely to happen is next year.