Volumes and Permutations

The factorial function n! = n \cdot (n-1) \ldots 3 \cdot 2 \cdot 1 is traditionally introduced as the number of permutations of n items.

It also crops up as the reciprocal of the (n dimensional) volume of the n dimensional simplex defined by

x_1 \textgreater 0, x_2 \textgreater 0, \ldots x_n \textgreater 0, x_1 + x_2 + \ldots + x_n \textless 1

There’s a routine proof by induction that the volume of an n simplex is 1/n!, but it leaves the connection with the number of permutations of n items a little mysterious.

Here’s one way to see why they’re related. Continue reading “Volumes and Permutations”

The Mobius Band Revisited

Here’s something I only recently realized: the mobius band arises naturally when you consider the space of all lines in the plane.

To begin with, the open mobius band (i.e. the usual mobius band without its boundary) is the same as the space of all lines in the plane. The easiest way to see that is Continue reading “The Mobius Band Revisited”

Complementary Independence

I was thinking through a probability situation the other day, and I noticed I was making an unspoken assumption, namely that if two events A and B are independent, then their complements \overline{A} and \overline{B} are independent.

At first I thought that this probably wasn’t true, because I’ve never seen that particular result explicitly stated in any probability or statistics book I’ve ever read (and I’ve read a few over the years), and because of the fact that human intuition for probability is notoriously unreliable (e.g. even Paul Erdos got the Monty Hall problem wrong when he first heard it).

Well, it turns out to be correct. In fact, a stronger result is true, namely Continue reading “Complementary Independence”