# Get under the table?

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was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

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The remnants of the Upper Atmosphere Research Satellite (UARS) are due to hit the earth later today and NASA have put up some details of their risk assessment. But this doesn't say where their '1 in 3200 chance of anyone being hit' comes from, and so can we get this figure from a back-of-an-envelope calculation?

The satellite has been up 20 years, stopped working in 2005, and weighs 5700 kg, about the size and weight of a double-decker bus. NASA say it will break into 26 objects that will survive re-entry, weighing 532 kg in total, about the weight of 8 washing machines. These will be spread over about 300 miles, but cover a total damage area of around 22 sq m (around 3 parking places).

The earth has surface area 500,000,000 sq km which is 500,000,000,000,000 sq m, and so assuming the bits can land anywhere, there is around a 1 in 20,000,000,000,000 chance of any particular square metre being hit. If You (that means *You*) make a target of say around 1 sq m, then assuming a random landing place there is around a 1 in 20,000,000,000,000 chance of You being hit - that's the same chance as flipping a coin 44 times in a row and it coming up heads every time. Or slightly better than the chance of winning the lottery twice in a row.

But there are 6,700,000,000 people on earth, if they each take 1 sq metre that’s 6,700 sq km, only 1/80,000 of the earth’s surface. So if everyone in the world went to Glastonbury Festival, they would only take up Somerset and Wiltshire combined, although the toilets can't be imagined.

So the chance of anybody at all being hit is 6,700,000,000 / 20,000,000,000,000 which is 1 in 3000, very close to NASA’s quoted figure of 1 in 3200.

In fact they have some idea where the debris will land (not in North America is all they seem to care about), but even with 2 hours to impact, they still can’t tell within 8000 miles where the bits will come down. So in fact this 1 in 3200 figure seems rather naive since the orbit is known and the population density underneath may make this chance bigger or smaller.

Should we get under the table? The largest object will weigh 158 kg [25 stone], about the weight of an adult gorilla, but that sounds a bit soft so perhaps better to think of a couple of washing machines tied together travelling at 100 mph. So no point in wearing a crash helmet.

If it's any consolation, bigger stuff came down last year, and nobody has ever been hit by space debris – yet.

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## Comments

Kimbo (not verified)

Fri, 23/09/2011 - 9:31am

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## maths

david

Sat, 24/09/2011 - 6:11pm

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## area of people

yes this is a good point, but the 24 square metres for the debris, and the 1 square meter per person already makes allowance for this overlap

stevek (not verified)

Fri, 23/09/2011 - 10:04am

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## " and nobody has ever been

Dave (not verified)

Fri, 23/09/2011 - 10:54am

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## Space Debris Victim

david

Sat, 24/09/2011 - 6:11pm

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## sorry, should have said

sorry, should have said 'killed by debris'

Rick Ansell (not verified)

Sat, 24/09/2011 - 10:52am

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## Impact Point

At the time the assessment was done the impact point was very poorly known. The orbital path crosses most of the earth’s surface on an interlaced path, missing only the poles. Until relatively recently even the month of re-entry was unknown so 'anywhere in that zone', biased a bit to the northern and southern extremes due to the maths of the path was the best they could do. So the number isn't 'rather naive' - it only 'seems rather naive' until you think about it and add a few facts.

The reasons the time was so uncertain has to do with the fact that the satellite is out of control, has been hit by debris and is known to be damaged, so its geometry isn't fully known and the way the atmosphere varys over time and geographic location.

david

Sat, 24/09/2011 - 6:14pm

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## yes maybe a uniform

yes maybe a uniform distribution for debris is reasonable in this case, although the absence of the (people-less) poles must lift the chances a bit. But I was surprised that the 'chance of hitting anyone' calculation appeared to be so back-of-envelope.