As you may or may not know, Taco Bell
has deployed a 40x40 ft
in the South Pacific
, and they have stated that if any piece of Mir
hits that target, everyone in the U.S.
gets a free taco
. I’m not sure how they’ll administer that, but I’m sure they’ll worry about this when it happens.
Some friends and I were laughing about this, and talking about the insurance policy they must have taken out on the occurrence. Then we started wondering what, say, Lloyds of London would do if approached with this promotion and asked for an insurance policy against the event happening. They’d need odds of it actually coming off! So how do you get those? One of my friends did some slacking at work, and I’ve shamelessly ripped off some of his numbers for this w/u, with some additions of my own.
According to the BBC, U.S. SPACECOM says the odds of any given
person being hit by Mir are 1 in 2 billion.
If a human occupies roughly two square feet when viewed from directly above (let’s not take grazing angles into this, it’s silly enough) then the target has 800 ‘human target sizes’ in it (40x40=1600; 1600/2 = 800).
So, let’s take 2 billion to one being the odds that a person will be hit, using the SPACECOM numbers. So, the odds that the target will be hit is:
Ptarget = Pperson * 800
Ptarget = 5 x 10-10 * 800 = 4 x 10-7
...or 1 in 2,500,000. So, using SPACECOM numbers and some SWAGs, we get slightly crappy odds of getting a free taco (if we live in the U.S.). Of course, as the BBC noted, this also would mean that roughly three people would get a severe bop on the noggin (6 billion people on the planet, 1 in 2 billion, you do the math).
Of course, let’s check that over using some better numbers. This in no way takes into account such wonders as the fact that they’re trying to steer Mir for a certain spot.
My friend came with the figure that Earth’s surface area is ~ 5*10^14 square meters. Assume that about 1000 different 1 square meter ‘plots’ are hit with significant
chunks. Assume these are independent (not true, but a start).
That gives 1 : 500,000,000,000 chance of any given square meter being hit, assuming a random reentry. If there are around 150 square meters of target (not exact, I know) then that means approximately 1 : 3.33 billion for the taco.
Now, let's say they think they can localize Mir pieces to an area of 500km x 200km. That means 1000 1 m2 plots would be smacked out of a total of 100 billion; that works out to about 1 : 100 million for the taco.
So, assume that ‘everyone in the U.S.’ means approximately 250,000,000 people. If those little greasewraps cost around a dollar each for TB to produce and sell (they don’t, but let’s keep it simple) then you’re looking at a potential cost of 250 million dollars. Well, if there is a one in 100 million chance of this cost of 250 million dollars coming due, the ‘common sense’ math would tell us that around $2.50 is the ‘expected payout.’
This is Lloyds, though. So they wouldn’t want to expose themselves (Ooo!) unnecessarily. I’m sure someone in the actuarial biz can make much better assumptions than I can, but I’d guesstimate that they wouldn’t be happy with less than around 1 million dollars. So, if Taco Bell were to actually buy this insurance policy, they’d be likely paying somewhere around 400,000 times the ‘actual expected payout’ from a stats point of view.