### Closed Under Means

Here's a nice little problem from the 13th All Soviet Union Mathematics Olympiad.
Given a set of real numbers $$S$$ containing 0 and 1 that's closed under finite means, show that it contains all rational numbers in the interval $$\left[0,1\right]$$.
This isn't a difficult problem, but there's a particularly nice solution.

First observe that if $$x\in S$$ then both $$\frac{x}{4}$$ and $$\frac{3x}{4}$$ are in $$S$$; average $$\{0,x\}$$ to get $$\frac{x}{2}$$, average $$\{0, \frac{x}{2}\}$$ to get $$\frac{x}{4}$$, average $$\{\frac{x}{2}, x\}$$ to get $$\frac{3x}{4}$$.

We could show any rational number $$\frac{m}{n}$$ with $$1\leq m < n$$ is in $$S$$ if we had $$n$$ distinct elements from $$S$$ that summed to $$m$$. Let's exhibit one.

Start with an array with $$m$$ 1s on the left, $$n-m$$ 0s on the right. Repeatedly replace adjacent $$x,y$$ values with $$\frac{3(x+y)}{4}, \frac{(x+y)}{4}$$, where either $$x=1,y\neq1$$ or $$x\neq 0, y=0$$, until there is one 0 and one 1 left. We can do this in exactly $$n-2$$ steps, each resulting number is guaranteed to be in $$S$$ by the above note, and each number is guaranteed to be distinct by construction!

Examples:

$$\frac{1}{3}: \left[1,0,0\right] \to \left[\frac{3}{4},\frac{1}{4},0\right]$$

$$\frac{2}{5}: \left[1,1,0,0,0\right] \to \left[1,\frac{3}{4},\frac{1}{4},0,0\right] \to \left[1,\frac{3}{4},\frac{3}{16},\frac{1}{16},0\right]$$

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD

New install:
Ubuntu 17.04
Titan V 12GB
/ partition on a 250GB Samsung 840 Pro SSD (had an extra around)
/home partition on a new 1TB Crucial MX500 SSD
New WD Blue 4TB HDD