## Probability and Statistics

Lecture notes‎ > ‎

## For today you should:

2. Do Homework 4.
3. Follow the instructions in Lecture 5 to generate a random string.

## Today:

1. Streaks and hot spots
2. Monty Hall
3. Nature of evidence

#### For next time:

1. Do Homework 5.

## Pick your favorite argument:1) Suppose there are a million doors and Monty opens all but yours and one other.2) Suppose that you decide to stick no matter what.  Then the switch talk is irrelevant and your chance of winning is 1/3.3) Apply Bayes's theorem (details)A little bit of history: Tierney's article in NYTFor the embarrassing list of letters from supposedly smart rude people, seehttp://www.marilynvossavant.com/articles/gameshow.htmlAnd what's the moral of that story?http://en.wikipedia.org/wiki/Cromwell's_ruleMonty Hall variantsUsing Bayes's Theorem to solve the Monty Hall problem is not necessarily easier than other ways, but it is easier to generalize.Suppose we change Monty's behavior:1) When Monty has a choice, he always chooses B.2) Monty chooses B or C at random, regardless of where the car is.  If he reveals the car, you lose.3) When Monty has a choice, he chooses B with probability p and C with probability 1-p.For the Blinky Monty problem, see http://allendowney.blogspot.com/2011/10/blinky-monty-problem.html

### The nature of evidence

What does it mean to say that E "is evidence for" H?

A reasonable definition is

P(H|E) > P(H)

Which implies that

P(E|H) / P(E) > 1

Or if there are just two hypotheses

P(E|H) > P(E|~H)

In other words, if the evidence is more likely under the hypothesis than under the alternative.

This is sometimes expressed by the likelihood ratio:

P(E|H)
----------   > 1
P(E|~H)

Two people have left traces of their own blood at the scene of a crime.  A suspect, Oliver, is tested and found to have type O blood.  The blood groups of the two traces are found to be of type O (a common type in the local population, having frequency 60%) and of type AB (a rare type, with frequency 1%).  Do these data (the blood types found at the scene) give evidence in favour [sic] of the proposition that Oliver was one of the two people whose blood was found at the scene?

Solution.  Don't peek!

The moral of this story is that evidence consistent with H is not always evidence in favor of H.