Category Archives: Maths

Consider a spherical, frictionless car of mass 1 …

A car of mass m drives at speed v towards a concrete wall. Its kinetic energy is ½ mv2, and all of that energy is dissipated when it hits the wall.

Now consider an identical car driving at speed 2v towards the same wall. Its kinetic energy is ½ m(2v)2 = ½ m4v2 = 2 mv— four times as much as the original car, since kinetic energy goes with the square of the velocity. I hope that up to this point, all this is uncontroversial.

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What is a good decision?

Suppose I come to you and offer to play the following game. I toss a fair coin. If it comes down heads, I give you £5; but if it comes down tails, you give me £10.

This is not a favourable gamble for you: the expectation is that I will win £2.50 — in other words, if we play this game n times, I will win a total amount that, as n increases, approaches £2.50 x n.

But suppose you just think, what the heck, let’s do it, and as it happens the coin comes down heads. You win £5 and walk away.

Now: when you decided to play, did you make a good decision?

Why armed guards in schools are a bad idea

I just saw this tweet from National Rifle Association (NRA):

On the assumption that this is a genuine query, I thought I’d take a moment to talk about some simple statistics and probabilities.

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First, Wikipedia notes that four presidents (Lincoln, Garfield, McKinley and Kennedy) have been shot by assassins. For simplicity, we will leave aside the failed assassination attempts on thirteen other presidents (and the failed attempts on the lives of Lincoln and Kennedy before the successful ones). Let’s consider the time from Lincoln’s death to now (147 years from 1865 to 2012), and say that the chance of a president being shot dead in any given year is 4 in 147, or about one in 40. (The real chance is surely much higher than that — note that there have been attempts on the lives of all the last eight presidents.)

The population of the US is 315 million, of which 27.3% are under 20 years of age. Let’s assume that about half of those are school age (between 5 and 15), which is 43 million schoolchildren. In 2012, there have been seven notable school shootings, but “only” 29 children murdered as a result. So let’s say that the chance of schoolchild being shot dead in any given year is 29 in 43 million, or about one in 1,500,000.

There were 600 accidental deaths by gunshot in the USA in 2010. Somewhere in the range of 30-34% of adults own a gun. Given that there are 230 million adults in the USA (and assuming that the number of children owning guns is negligible), that means there are about 74 million gun owners in the USA. So the chance of any gun owner accidentally killing someone in a given year is 600 in 74 million, or about one in 123,000.

In reality, of course the armed guards who protect the president are the best of the best: very highly trained, and much less like to have accidents than the general gun-owning population. But even assuming they are no more competent than hypothetical armed school guards, here’s how it works out.

  • Giving the president an armed guard increases his chance of being shot, due to accident, by one in 123,000. Given that his chance of being shot is already one in 40, this is negligible.
  • Giving children an armed guard increases their chance of being shot, due to accident, by one in 123,000. Given that their chance of being shot was previously one in 15,000,000, it means they are now 122 times as likely to be shot.

These numbers are all approximate. I could easily be wrong by a factor of two or more. Even if I’m wrong by a factor of six, it still means that the president is much, much, much better off with an armed guard where as a schoolchild would be twenty times as likely to be shot.

I hope that clears things up.