Math 300 , Spring 2008, M Feb. 11, Day 7 after class Hit reload to get most current version

See Day 6 for HW, Reading,  notes on Conditional probability.
Next: Bayes' theorem Ash pp 58-61, M&M pp. 321-22.
Then Independence, M&M pp. 266-70, 312(rule5), 322.  Ash pp.41-44
HW, Conditional probability, any remaining from Day 6.  Also these:
 Assigned Day 6, on Back of Decision Analysis, #4.96(surgery or med management)  Note, the tree is only for finding P(A) under the surgery choice.
--Moore&McCabe 5th ed. p. 331 4.125  Do this problem with x as the actual proportion  of plagiarism, y as the proportion who answer No.  Solve for x in terms of y.

The time-blindness of Lady Luck: Simplify each probablity to a simple fraction.
A.  Suppose one card is dealt to each of 3 people and you are the third person.  What is the probability that YOU get a heart?  Do it with a tree: first card, second card, third card is heart or not.

B. Today a small grocery store has 6 cartons of milk, 2 of which are sour.
a) If you are going to buy the 4th carton of milk sold today at random, compute the probability that your carton is sour.  Do it by making a tree; first carton sold Sour or Not, 2nd Sour or Not, etc. up to your 4th.
b) If you buy the first carton of milk today, what is the probability that it is sour?

(Note, the final  probabilities for A and Ba were calculated by being conditioned on the past, but turn out to be the same as if they were "decided" first.   So my getting a heart IF person one or two already has, does depend on those previous events.  But my getting a heart considered by itself alone, does not.  Our intuition tends to "know" that the past matters, wrongly sometimes.)

Postpone :
First Bayes' problems.  Finish Handout: "Conditional Probabilities (Bayes)" #3 (Death sentence) and do #2 (handedness)

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Recap: Conditional probability and chain rules:  (Ash 2-1, 2-4 to p. 58)
             P(A | B) = P (A and B)
                              P( B)
3 presentations:  Venn diagram (this privileges sets over complements)
 Two-way table
Tree (especially good for causal or decision sequence; can work for sequence of 2, 3, or more things).   Written out in formal notation, "chain rule" or "multiplication rule."   P(A and B) = P(B)P(A|B) 

There is often a confusion in textbooks between the conditional probability  as:
 (1)--Something calculated from a known Sample Space or probability model:  partial knowledge is obtained (A is true, has occurred, is perceived) which changes probabilities assigned to other events.
(2)--The formula used to construct a probability model when the model has a natural causal or decision chain, the probabilities at each step are known given the previous step, and the task is to construct probabilities for events at the end of the chain especially (and maybe for the intermediate events also.).

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If our probabilities are in a table, it's as easy to find P(S|W) as it is to find P(W|S).
But if  we built our model using a tree, it may seem hard to get a conditional probability "back" from the last thing we see to an earlier thing.

Bayes' theorem:  Don't memorize the theorem, just know the process.
From knowing an outcome W, find the probability of an antecedent S.
       P(S | W) when W|S is the direction you have information for.
These can always be modeled effectively with a tree, S on the first branching, W on the second:
    Find the probability of the outcome W by summing the "fav" tree path results. (Ash's "total probability" p. 58)
    Note the probability of (S and W): just the single path with S followed by W.
    Divide: (SandW path) / (sum of W paths)
              = P(S and W)/P(W) = P(S|W)

Handout Death penalty (#3)