Million-to-one shots come up nine times out of ten.

- Terry Pratchett

While researching rules for ninjas to live by (don't ask), I came across a mathematical phenomenon I'd never noticed noticing. It's this:

**The Law of Inverse Ninjas:** The probability of a group of ninjas winning a battle in a film is inversely proportional to how many of them there are.

Which means: if you're faced by a gang of 1,000 ninjas, you probably shouldn't worry too much, because they'll attack you one at a time in a blasé way; you can pick them off and destroy their morale.

However, if it's you and your buddies against a lone sword-wielder, you'd better run away scared: that's when the ninja is at his or her most powerful.

### This is SPARTA! (and inverse proportion)

In this context, it makes the Spartans' victory at Thermopylae all the less impressive. The probability of the Spartans winning with 300 soldiers would be proportional to $\frac{1}{300}$ - so let's call that $S = \frac{k}{300}$. The probability of the Persians winning with their 100,000-strong army would be $P = \frac{k}{100,000}$. And we know $S + P = 1$, because you don't draw battles.

Then : $\frac{k}{300} + \frac{k}{100,000} = 1$

Multiply by 300,000: $1000k + 3k = 300,000$

So $k = \frac{300,000}{1003}$

(If you've been following the ninja maths series, you'll know that's about 300 (-0.3%), or 299.1)

So, the odds on Sparta winning were 299.1 out of 300, or 99.7%.

### Kill Bill, the Spanish Conquest of Peru and Short-handed Football

In fact, it turns out that under the Law of Inverse Ninjas, your chances of winning are (people against you) / (total people in the battle). So, when Kiddo takes on the Crazy 88 squad, she's got an $\frac{88}{89}$ chance of winning, slightly less than 99%. Against Bill, she was back down to 50-50.

When Pizarro and his army of 200 or so Spaniards pitched up in Cajamarca, Atahualpa made the elementary mistake of showing up with 80,000 troops rather than on his own, turning a 99.5% chance of victory into about 0.25%.

And the old football adage about a football side being more dangerous when they're down to ten men? That's just the Law of Inverse Ninjas in action. You'll find they often score the winner in ninjary time, which is a whole other story.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## SandiAmorim

I didn\’t understand a lot of this, but I LOVE the first part.

Going to look for my ninja outfit….

## Cav

I recall an episode of sharpe where Sean Bean asked one of his men, “how many of them are there,” got the reply “thousands” and stated “I like them odds.”

## Cav

He must’ve been familiar with the law!