Brad Efron wrote to me and posed an interesting statistical question:
“Last Wednesday Diane Sawyer interviewed an Oklahoma woman who twice
had had her home destroyed by a force-4 tornado. “A one in a
hundred-trillion chance!” said Diane. ABC showed a nice map with the
current storm’s track of destruction shaded in, about 18 miles long
and 1 mile wide. Then the track of the 1999 storm was superimposed,
about the same dimensions, the two intersecting in a roughly 1 square
mile lozenge. Diane added that the woman “lives right in the center of
Tornado alley.”
Question: what odds should have Diane quoted? (and for that matter,
what is the right event to consider?)
Regards, Brad”
Anyone have a good answer?
By the way, I should add that Diane Sawyer has a history of
broadcasting stories filled with numerical illiteracy. She did a long
series opposing the use of lean finely textured beef (LFTB), also
known as “pink slime.” In fact, LFTB is perfectly healthy, its use
requires slaughtering many fewer cows each year and makes meat cheaper
for poor people. The series was denounced by many scientists and even
environmentalists. ABC is being sued for over one billion dollars.
She also did a long series on “Buy America” encouraging people to
shun cheap goods from abroad. This is like telling people who live in
Cleveland to shun buying any products and services not produced in
Cleveland (including not watching ABC news which is produced in New
York, or reading statistics papers not written in Cleveland.) This
high-school level mistake in economics is another example of Ms.
Sawyer’s numerical illiteracy.
But I digress.
Let’s return to Brad’s question:
What is a good way to compute the odds that someone has their house
destroyed by a tornado twice?
I open it up for discussion.
Normal Deviate 
12 Comments
For any location in the US, there are very detailed data showing the history of tornado frequency, strength, path, and time of year. So it should be pretty straightforward to compute the frequency of tornadoes of a given strength in a given location. See http://www.spc.noaa.gov/exper/envbrowser/ for a nice visualization of the data.
Let S be some set of people, and let property U be unluckiness in the form of having experienced at least two home destructions due to F4 tornadoes. Here are two events that might be of interest: (1) There exists a person in S with property U; (2) a person selected uniformly at random from set S has property U.
‘She also did a long series on “Buy America” encouraging people to shun cheap goods from abroad. This is like telling people who live in Cleveland to shun buying any products and services not produced in Cleveland.’
I disagree. I think it makes more sense to view Sawyer’s series as an effort to deliver a positive shock to net exports. Valid criticism might be made on this basis, but not on the basis that promoting intra-country trade is very similar to promoting intra-city trade. After all, the U.S.A. is a fiscal union in which internal barriers to trade are low and worker mobility is high, permitting a lot of local specialization.
I would do a simulation for an accurate estimation on this one but, for the purpose to check how right Ms. Diane Sawyer is with her “one in a hundred trillion chance”, I would calculate a minimum value by considering the total number of houses in the highest tornado risk area in the US and the number of houses that were destroyed twice in that area.
The true odds will be higher than this calculation, but this one is simple to do and my guess is that it will be way, way bigger than Ms. Sawyer estimation and enough to prove her way, way wrong… which seems the whole point of the calculation in this case.
Gee.. I mixed up higher and lower everywhere! It should be the houses destroyed twice in the lower risk area among the tornado areas… I really need to read twice before I shoot Enter.
In fact, we can forget about the lower risk area, using all the tornado areas already gives a minimum value estimate so, even easier.
Here’s an order-of-magnitude estimate using a very crude Poisson model, suitable for my freshman undergrads.
Let the estimated annual rate of destruction be the ratio of destroyed houses in Oklahoma to total houses in Oklahoma (a la Banana’s comment). Then find p = Pr(D > 1 | rate*t) where D ~ Poisson(rate*t). Remember, this is like the Birthday Problem: any house, just like any day of the year, will do for the improbable TwoFer. For example, t=15 years, and if rate = 0.001, then p ~= 1.1×10^(-4), about 1 in 9000; rate = 0.0001 gives p ~= 1.1×10^(-6), about 1 in 900,000.
So Sawyer would have been off by anywhere from 8 to 10 orders of magnitude. No wonder we think journalists are clueless gits.
Given that you know the tracks for a number of years, a more accurate estimate can be obtained by dividing the area into house size blocks and then seeing how often the tornadoes cross. Then modify this probability based on number of blocks occupied by houses.
In many ways I find this type of question a bit pointless. After all, it happened, so it has probability one of happening. Chances of it happening again are probably very small.
This page, written by a meteorologist, while a little outdated, has some musings on possible approaches to the calculation, and more interestingly, also lists some references on the distributions of violent tornados http://www.flame.org/~cdoswell/tor_probs/vtornado_prob.html
I’m guessing Diane Sawyer would benefit from watching this old gem: http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool_juries.html
This is a similar to “Prosecuter’s Fallacy” where lawyers purposely bamboozle jurors with false probability statistics.
They treat two such conditional events as wholly independent events and/or falsify conditional probability so that P(A/B) is the same as P(B/A).
OJ Simpson’s defence team admitted such falsification tactics. Lawyers are not on oath to tell the truth.
Sally Clark, a UK mother, who was convicted for 2 cot deaths was convicted on this false premise from an “expert” witness.
He stated a 7 million to one chance that she had not murdered both her children.
The true mathematical value was that two such cot deaths are 9 times more likely than two such murders.
The correct event question for the tornados is:
if a tornado has already wrecked a home what is the probability that a second tornado will also wreck the rebuilt home.
It needs local and physical data to solve and depends on whether the second home was reinforced to withstand the first tornado force or more.
I once met a man who told the story that once, long ago, he was to take a plane, but there was an overbooking problem. There was a bit of heavy arguing, and finally he was left . Then he stood watching he plane approaching the horizon — and the going down in flame! No one survived. People find this sort of thing very unlikely! Then I like to point out (and most people understand it) that while such things are indeed unlikely to happen tou YOU, TOMORROW, it is not at all unlikely that such things happen, somewhere, someplace, to somebody.
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