let

1

• lets talk about conditional probability by
  considering a specific example
• suppose we roll a pair of dice and are interested
  in the probability of getting an 8 or more (sum
  of the spots gt 8). what is the unconditional
  probability of this happening?
• now what if when I roll the dice, one of them
  rolls under the chair, and all I can see is the
  other die with 5 spots on the up-face. what is
  the conditional probability that the sum of the
  spots gt 8 given that one of the dice has 5
  spots?
• notice that knowledge of the one dies 5 spots
  essentially changes the sample space from S of 36
  points to one of just 6 points (5,1), (5,2),
  (5,3), (5,4), (5,5), (5,6) and so the probability
  should be 4/62/3
• note that this coincides with the definition of
  conditional probability given on page 80
• since P((5,3),(5,4),(5,5),(5,6) /
  P((5,1),(5,2),(5,3),(5,4),(5,5),(5,6))
• (4/36) / (6/36) 4/6 2/3
• we usually use this relationship to compute
  probabilities of non-independent events

2

• and then we define two events to be independent
  whenever
• then we get the usual formula for and for
  independent events
• go over the water quality example on page 83.
  this example assumes that successive water
  samples are independent of each other...
• a commonly used application of conditional
  probability is given in Bayes Theorem (page 87)
  but first lets look at a preliminary result
  called the theorem of total probabilty via the
  example at the top of page 85
• suppose a plant gets part VR from one of three
  suppliers (B1, B2, B3) 60 of all VRs come from
  B1, 30 come from B2, and 10 come from B3. the
  three suppliers have varying records as to the
  quality of their product (95, 80, 65) perform
  as specified. Choose a VR at random what is
  the probability that it performs as specified?
• let AVR performs as specified. then show the
  total probability of A as broken up into its
  parts as determined by the three suppliers (do
  a Venn diagram and a tree to show how this
  works...)

3

• now consider a related problem that can be solved
  by Bayes Theorem what is the probability that a
  randomly chosen VR is from supplier B1 given that
  it performs to specifications? Notice that this
  is the reverse of the probabilities in the
  theorem of total probabilities...so first write
• then rewrite the denominator in terms of the
  theorem of total probability and we have Bayes
  Theorem (Theorem 3.11 on page 87)
• note that the numerator of Bayes Theorem is the
  probability of A going through the rth branch of
  the tree and the denominator is the sum of the
  probabilities of A going through all the branches
  of the tree... see the rest of the authors notes
  on this theorem on page 87.
• go over the example at the bottom of page 87
• HW Read 3.6 and 3.7 and do the following
  problems3.64, 3.66, 3.67, 3.69, 3.71-3.75, 3.79