Math 300 , Spring 2004, W Day 8 Hit reload to get most current version

See Day 7 for HW, Reading,
  Three models of probability.

Conditional probability and chain rules:  (Ash 2-1, 2-4 to p. 58)
             P(A | B) = P (A and B)
                              P( B)
Prob of A given B =  Prob of (A and B)  divided by Prob of B
A Given B: B is the whole thing now!
      Throw away the part of A not in B:  P(A and B)
      Divide by P(B): standardizes (A and B) as a proportion of the "whole" B.
3 presentations:  Venn diagram (this privileges sets over complements)
 Two-way table (Table expands to 2 dimensions (e.g. wind absent, light, heavy), doesn't privilege anything),

	S	Sc	total	Bold numbers are basic numbers from Venn  diagram.  All other numbers follow.   P(W|S) = W &S cell    divided by S column total.
W	.3	.2	.5
Wc	.1	.4	.5
total	.4	.6	1.0

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."
Suppose we know P(S) = .4 = 2/5 , and  that Snow or not produces different probabilities of Wind:
 P(W|S) = 3/4, P(W|S^c) = 1/3.
 

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.).

Philosophically, these are different.   However, they are consistent.  If you use chaining or a tree to construct a model, and then calculate the conditional probabilities for pieces in the tree from your total sample space, they will come out with the same values as the ones you used in constructing the tree. (Whew!)  I think it's helpful to keep track of whether you're doing (1) or (2) at any given moment.

Example 3, p. 40: Chain rule (Multiplication rule) represented as tree.

Read ahead?:   Bayes' theorem Ash pp 58-61, M&M pp. 349-50.
Then Independence, M&M pp. 294-296, 340, 350.  Ash pp.41-44