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
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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
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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