# FA17 Lecture 5

## Contents

# Lecture 5: conditional probability

- Reading: [[../handouts/cameron_prob_notes.pdf#page=31|Cameron chapter 2]]
- [[../../2017sp/lectures/lec13-conditioning.html|Last semester's notes]]
- Definitions: conditional probability
- Modeling with conditional probability
- Bayes's rule, Law of total probability

## Conditional probability

**Definition:** If and are events, then the **probability of A given B**, written is given by Note that is only defined if .

Intuitively,

is the probability of in a new sample space created by restricting our attention to the subset of the sample space where occurs. We divide by so that .**Note:** is not defined, only ; this is an abuse of notation, but is standard.

## Probability trees

Using conditional probability, we can draw a tree to help discover the probabilities of various events. Each branch of the tree partitions part of the sample space into smaller parts.

For example: suppose that it rains with probability 30%. Suppose that when it rains, I bring my umbrella 3/4 of the time, while if it is not raining, I bring my umbrella with probability 1/10. Given that I bring my umbrella, what is the probability that it is raining?

One way to model this problem is with the sample space

= \{(r,u), (nr,u), (r,nu), (nr,nu)\}[/math]

Let

be the event "it is raining". Then . Let be the event "I bring my umbrella". Then .The problem tells us that

. It also states that while . We can use the following fact:**Fact:** . Proof left as exercise.

to conclude that

and .We can draw a tree:

We can compute the probabilities of the events at the leaves by multiplying along the paths. For example,

To answer our question, we are interested in

. We know . We can compute using the third axiom; . We can then plug this in to the above formula to find .Note we could also answer this using Bayes's rule and the law of total probability (see below); it would amount to exactly the same calculation. The tree just helps organize all of the variables.

## Bayes's Rule

Bayes's rule is a simple way to compute

from .**Claim:** (Bayes's rule): .

**Proof:** Write down the definitions; proof left as exercise.

## Law of total probability

**Claim:** (law of total probability) If partition the sample space (that is, if for and ), then

**Proof sketch:** Write . Apply third axiom to conclude . Apply definition of .