Probability%202

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Probability 2

• Professor Jim Ritcey
• EE 416
• Please elaborate with your own sketches

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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.

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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

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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

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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?

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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!

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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

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Classic Vendor Example
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Classic Vendor Example
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Classic Vendor Example