# Segment 1. Let's Talk about Probability

## Contents

#### Watch this segment

The direct YouTube link is http://youtu.be/H5WjVgL6Nh4

Links to the slides: PDF file or PowerPoint file

#### Bill's comments on this segment

Well, I do sound nervous! This was one of my first webcasts. The production values get a little better with later segments. However, the material here is important, so be sure you understand it before going on.

Here is a link to the paper by R.T. Cox, discussed on slide 2. It's surprisingly readable for something so fundamental.

### Problems

#### To Calculate

1. Prove that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(ABC) = P(B)P(C|B)P(A|BC)}**
.

2. What is the probability that the sum of two dice is odd with neither being a 4?

#### To Think About

1. First-order logic is a type of propositional calculus with propositions **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c}**
and quantifier symbols **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists}**
. This allows statements like "Socrates is a philosopher", "Socrates is a man", "There exists a philosopher who is not a man", etc. Can you use first-order logic as a calculus of inference? Is it the same as using the probability axioms? If not, then which of Cox's suppositions is violated?

2. You are an oracle that, when asked, says "yes" with probability **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}**
and "no" with probability **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-P}**
. How do you do this using only a fair, two-sided coin?
As we did in class. Represent P as a binary number. Whenever

3. For the trout/minnow problem, what if you want to know the probability that the Nth fish caught is a trout, for N=1,2,3,... What is an efficient way to set up this calculation? (Hint: If you ever learned the word "Markov", this might be a good time to remember it!)