FA17 Lecture 5
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
Definition: Ifand 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.
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
Letbe 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 thatand .
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 is a simple way to computefrom .
Claim: (Bayes's rule):.
Proof: Write down the definitions; proof left as exercise.
Law of total probability
Claim: (law of total probability) Ifpartition the sample space (that is, if for and ), then
Proof sketch: Write. Apply third axiom to conclude . Apply definition of .