The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

My Photo
Location: Chandler, Arizona, United States

Monday, December 01, 2008

The 100 Year Storm

We sometimes hear phrases like "a 100 year storm or flood" or a "500 year storm or flood" or a "once in 10,000 year asteroid strike" used and most understand them to mean that such events will occur only once every 100 or 500 or 10,000 years. This is a misinterpretation of the statistics involved. The probability of a certain number of events in a certain time period can be calculated using the Binomial Theorem. Assume the annual probability of an event is 'p', so that the probability of it not occurring is (1-p). Then evaluate the following expression using the Binomial Theorem:
    [p + (1-p)]n = ∑ni=0 n!/[i!·(n-i)!] pi·(1-p)(n-i)
Each term represents the probability that exactly i events will occur in the n year period.

As a concrete example, consider a 100 year flood and a time period of 100 years. The annual probability of such an event is 0.01, and we take n = 100 for a 100 year period. Evaluating the first few terms of the Binomial Series:

i n!/[i!·(n-i)!] pi·(1-p)(n-i) E
0 0.366032341 0
1 0.369729637 0.369729637
2 0.184864818 0.369729637
3 0.060999165 0.182997497
4 0.014941714 0.059766859
5 0.002897787 0.014488935
6 0.00046345 0.002780704
7 0.000062863 0.000440044
8 0.000007381 0.000059053
... ... ...
Sum 0.999999156 0.999992366
The second column is the probability that exactly i events occur in a 100 year period, and the third column is the Expectation, which is the product of the numbers in the first two columns. As expected, the probabilities add up to 1 (neglecting the small probabilities for i > 8), and the expectation also adds up to one (again, neglecting the omitted terms), so we would expect an average of one event over many 100 year periods.

What is counter-intuitive is that there is almost as great a chance that no events occur (36.6%) as there is that 1 event occurs (37.0%), and since:
    1 - 0.366032341 - 0.369729637 = 0.264238022
there is a 26.4% chance that 2 or more events will occur in any 100 year period. It is therefore more likely that no events or 2 or more events will occur (63.0%) than that only 1 event will occur, a result most people would probably not believe !

Of course, this calculation assumes that the events from year to year are independent, but since weather seems to follow multiyear cycles, the independence from year to year is doubtful, and it seems likely that the probability of multiple events is higher than this calculation indicates.
P.S. - For those unable or unwilling to make such calculations, here is an on-line binomial calculator.

Dammit, Janet, The One Loves You **

One of the few good aspects for Arizona of the One's election is he is ridding us of Gov. Janet Napolitano. Her administration has been a disaster for the State, and she is skipping town before having to deal with a $1.2 billion deficit caused by her inability to deal with the Republican Legislature. EspressoPundit has written a series of blog posts concerning Napolitano's failures.

First, for your consideration, is her supposed accomplishments as the "Education Governor" in The Sanctification Begins...:

Napolitano advanced cause of education
If, as expected, Janet Napolitano, the ''Education Governor,'' joins the Obama administration, some Arizona education leaders worry if they will keep the important gains made under her leadership. ... Actually, Napolitano's real legacy with respect to education is that Arizonans can no longer afford to attend college. ... Despite her efforts, if Napolitano heads to Washington, D.C., she will leave the state still nearly last in the country for K-12 funding, with a high-school graduation rate of 70 percent and state universities burdened with students unprepared for college work. ... Her real legacy is that she made no progress on the [Teacher] Union's key issues but succeeded in making College unaffordable despite a provision in the Arizona Constitution that says a college education should be "As nearly free as possible."
Read the entire thing and note the graph of college tuition costs with its abrupt increase in slope under Gov. Janet's leadership, a fact confirmed and updated in the East Valley Tribune article ASU Seeks Tuition Hike

Second, for your consideration, is a review of her relationship with the Legislature, Women in Leadership:

“Napolitano’s team didn’t understand how the Legislature works, which is why bills didn’t get through the Legislature and she ended up vetoing so many bills that reached her desk,” Patterson said. “The state is out of money and out of gimmicks. The only thing left is to cut programs.” ... The governor has shown a complete disregard for these complex traditions and has a dismal relationship with the legislature. After six years in office, her legislative agenda is often dead on arrival and she famously refuses to discuss legislation until it ends up on her desk. Not surprisingly that legislation often contains features she considers objectionable and she's "forced" to veto scores of bills. Her string of vetoes is not a sign of strength. It is an indication that she and her team don't really understand the legislative process.

Third, for your consideration, is the prospect of her legacy in Nasty Brutish and Short:

The Napolitano administration appears to be winding down and Arizona Democrats have moved from denial to anger. I've been getting feedback from rank and file Democrats and they are unbelievably ticked off that Gov. Napolitano may resign and leave Republicans in charge of state government. ... Her spending programs--all-day kindergarten, Biotech, shinny university buildings -- were an illusion built on bubble revenue. Her refusal to trim spending when the market went south means that the eventual cuts will be drastic. Moderate Republicans who enabled her 2004 budget shenanigans were wiped out in the 2004 election and the moderate Republicans who enabled her 2008 budget shenanigans were wiped out in the 2008 election.
    Democrats have awakened to the fact that the state is bankrupt and the Republican majorities in the legislature are larger and much more conservative than when she took office.
    Perhaps more important than the trail of failed programs and busted budgets that she's leaving behind, is the fact that she's wiping out the Democratic farm team. Hundreds of high-level state Democrats owed their high-profile jobs to Napolitano's occupation of the ninth floor. The heavy handedness that she exerted as governor tainted her appointees and staff as well, and now many of them are unemployable.

Perhaps the other benefit of the election will be that Arizonans realize that it is time to retire "Johnnie Boy" Maverick McCain -- that Campaign finance reform was an unconstitutional disaster; the Gang of 14, institutionalizing the filibuster of judicial nominees, was a huge mistake; that McCain-Lieberman carbon cap-and-trade would ruin the entire economy; and that McCain's call for a "surge" in 2005 was premature and would have failed since there were not the other elements in place to allow it to succeed at that early date. There is little doubt he ran one of the worst campaigns in recent history, with the possible exception of Bob Dole. I'm convinced there is something about the desert sun here in Arizona which adles old conservative brains, e.g., Barry Goldwater, Sandra Day O'Connor, and Johnnie Boy.

**  A reference to the song Dammit, Janet from the Rocky Horror Picture Show.
UPDATE: EspressoPundit weights in with an update on Napolitano staff's future in What Color is Your Parachute ?:
Here's the list of the Governor's staff and it's really striking...well, appalling is probably the better word. For one thing, there are 150 people on the list and the total salary is $8.3 million. Dude, when did the Governor get an entourage like that?
In fact, the severance cost alone for the entire staff amounts to over $870,000!
Robert Robb at the Arizona Republic discusses The Napolitano Legacy:
Napolitano remains highly popular, even though a gaping budget hole has developed on her watch. Substantively, however, Napolitano's record is much thinner and illustratively so.

In 2002, Napolitano ran on a blistering critique of Arizona's status quo. Arizona ranked near the bottom on educational funding and achievement. Student-teacher ratios were too high. Dropout rates were among the highest in the nation. We had a low-wage economy that left the state in the cellar on per-capita income.

After six years of her governorship, none of these measures have materially moved.
And finally, over at The Sonoran Alliance, they are discussing one of Janet's real legacies, What is Janet Napolitano’s legacy? BIG BROTHER STATE OF SPEED CAMERAS:
Websites are popping up around the Arizona blogosphere and everyone from the Republic to the New Times has been covering the growing opposition to the cameras, which were rammed down our throats by Napolitano. The Republic ran an editorial today listing some of the problems with speed cameras and noting that even its supporters are now admitting the cameras have gone too far, they’re so prevalent people are disturbed.
All in the name of what? Greed. Not public safety, since drivers don’t get points on their record for these types of tickets, they can get 100 speeding tickets by photo radar and nothing will happen to their driving privileges. Further, a study in Texas found that accidents increased by 52% after speed cameras were installed! And the money doesn’t even go to improve the roads - it goes to fund Clean Elections!

News sources are also pointing out that the speed cameras aren’t going to bring in the revenues claimed; because 60% of drivers appeal the tickets, it will clog the justice courts, at an increasing cost to taxpayers.