# Segment 2. Bayes

## Contents

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Links to the slides: PDF file or PowerPoint file

Here is a link to the Efron paper mentioned.

At 27:28, you say that "in world 2, a tenth of all the birds are black crows", but it looks a lot more like $\frac{200\,000}{1\,000\,000}=\frac{2}{10}$ to me. --Noah 22:01, 17 January 2013 (CST)

I think so too. --Swang

There are 200,000 black crows and 1,800,000 white crows. --Jhussmann 09:29, 18 January 2013 (CST)

--- For "The Think About Question 2", are we trying to find the probability of the Knight crossing safely when he finds a troll? Or are we finding the probability of the knight crossing safely regardless of what he finds? - LoriL

### Problems

#### To Calculate

1. If the knight had captured a Gnome instead of a Troll, what would his chances be of crossing safely?

2. Suppose that we have two identical boxes, A and B. A contains 5 red balls and 3 blue balls. B contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box. What is the probability that it is blue? If it is blue, what is the probability that it came from box B?

#### To Think About

1. Do you think that the human brain's intuitive "inference engine" obeys the commutativity and associativity of evidence? For example, are we more likely to be swayed by recent, rather than older, evidence? How can evolution get this wrong if the mathematical formulation is correct?

2. How would you simulate the Knight/Troll/Gnome problem on a computer, so that you could run it 100,000 times and see if the Knights probability of crossing safely converges to 1/3?

3. Since different observers have different background information, isn't Bayesian inference useless for making social decisions (like what to do about climate change, for example)? How can there ever be any consensus on probabilities that are fundamentally subjective?