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Probability%202

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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 ... – PowerPoint PPT presentation

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Title: Probability%202


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( BA) P(AB)/P(A) only when P(A) gt0
  • 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(BA)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(AB) P(A) P(BA) 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 lt F
  • We can ignore Alf and the sample space is
  • Outcomes BltF, FltB 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 AltBltF
  • Outcomes ABF, AFB, FAB, BAF, BFA, FBA equally
    likely 1/6 by insufficient reason.
  • Then B ltF 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 ltF has
    occurred. Find P( BltFAltF )? Enumerate!

7
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8
Bayes Rule
  • Bayes Rule is simply the equality derived by
  • P(AB)P(B) P(AB) P(BA) P(BA)P(A)
  • Or P(AB) P(BA)P(A)/P(B), P(B)gt0, P(A)gt0
  • Critical tool for inference working backwards
    by induction

9
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Classic Vendor Example
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Classic Vendor Example
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Classic Vendor Example
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