Probability 2

1. 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 provides no new information as it leaves the probability unchanged

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!

8. 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 – working backwards by induction

11. Classic Vendor Example

12. Classic Vendor Example

13. Classic Vendor Example