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

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.

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

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)

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?

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!

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

Note that B must be contained in union A_n

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)

Classic Vendor Example

Classic Vendor Example

Classic Vendor Example