The dot product is a property of real vector spaces which is a simplified version of an inner product; specifically, it obviates the need to complex-conjugate anything because, well, \(\bar{n} = n, n \in \mathbb{R}\). The dot-product also yield a real number.

constituents

• \(x, y \in \mathbb{R}^{n}\) (NOTE the realness)
  • where, \(x = (x_1, \dots, x_{n})\) and \(y = (y_1, …, y_{n})\)

requirements

As we are familiar with, element-wise product and sum

\begin{equation} x\cdot y = x_1y_1 + \dots + x_{n}y_{n} \end{equation}

One envelope has 10 times the money in the other money.

WLOG let \(x\) be the envelope in Cary’s hand. The money in \(y\), then, \(y = \frac{1}{2}\qty(\frac{1}{10}x)+\frac{1}{2}\qty (10x) = 0.05x+5x = 5.05x\). Wat.

Basically; regardless if Cary took the envelope \(x\) or \(y\), the other envelope is expected to have \(5\times\) more money. What.

Why?

There’s a bug in this:

\begin{equation} y = \frac{1}{2}\qty(\frac{1}{10}x)+\frac{1}{2}\qty (10x) \end{equation}

is not true! There is a human PRIOR BELIEF!! Its very unlikely that mykel/chris put 10000 dollars into an envelope; so each individual amount in an envelope has an exogenous probability of it happening!

A human gene similar to the gene PreTA found in E. Coli, a bacterial found in microbiome. See effects of PreTA on Fluoropyrimidine, and by proxy Capecitabmine for implications on cancer treatment.