1. Professor Jim Ritcey EE 416 Please elaborate with your own sketches
Probability 2
Professor Jim Ritcey
EE 416
Please elaborate with your own sketches

2. Disclaimer
These notes are not complete, but they should help in organizing the class flow.
Please augment these notes with your own sketches and math. You need to actively participate.
It is virtually impossible to learn this from a verbal description or these ppt bullet points. You must create your own illustrations and actively solve problems.

3. Conditional Probabilities
Given (S, E, P) with events E ={ A,B,C, … }
Pick two events A and B. Define
P( B|A) = P(AB)/P(A) only when P(A) >0
This is the conditional probability of B given A
Draw a picture using Venn diagrams!
It is often easier to remember that
P(AB) = P(B|A)P(A),
Recall that AB = A cap B , the intersection

4. Conditional Prob & Independence
2 events are independent when P(AB) = P(A) P(B)
Under independence
P(A|B) = P(A) & P(B|A) = P(B)
Under independence, the condition (given B) provides no new information as it leaves the probability unchanged P(A|B) = P(A)

5. Ranking Example (MacKay)
Fred has two brothers Alf and Bob.
What is the probability that Fred is older than Bob { B < F }
We can ignore Alf and the sample space is
Outcomes { B<F, F<B } equally likely ½ by insufficient reason to assume otherwise
But what if we include Alf?

6. Ranking Example (MacKay)
We can include Alf and the sample space is
All 3!=6 rankings of A,B,F. Write ABF = A<B<F
Outcomes { ABF, AFB, FAB, BAF, BFA, FBA } equally likely 1/6 by insufficient reason.
Then {B <F} = {ABF,BAF,BFA} = 3/6 =1/2
Read { a,b,c} = {a} OR {b} OR {c}
Now Fred says he is older than Alf {A <F} has occurred. Find P( B<F|A<F )? Enumerate!

7. Ranking Example (MacKay)
The condition that Fred is older than Alf
Excludes some outcomes in the event of interest
Conditioning refines our knowledge

8. Note that B must be contained in union A_n

9. Bayes Rule Bayes Rule is simply the equality derived by
P(A|B)P(B) = P(AB) = P(BA) = P(B|A)P(A)
Or P(A|B) = P(B|A)P(A)/P(B), & P(B)>0, P(A)>0
Critical tool for inference
Given P(Output|Input) and observation of an output, compute the likely Input
P(Input|Output) = P(Output|Input) X P(Input)/P(Output)

12. Classic Vendor Example

13. Classic Vendor Example

14. Classic Vendor Example