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let’s talk about conditional probability by considering a specific example:

let’s 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 >= 8). what is the unconditional probability of this happening?

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let’s talk about conditional probability by considering a specific example:

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  1. let’s 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 >= 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 >= 8 given that one of the dice has 5 spots? • notice that knowledge of the one die’s 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 4/6=2/3 • 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 let’s 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 A=“VR 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 B1given 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 author’s 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 problems:3.64, 3.66, 3.67, 3.69, 3.71-3.75, 3.79

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