Math 300 , Spring 2008, W Feb 13 Day 8 Hit reload to get most current version

HW on Bayes: Draw trees, compute.  Make a table if you think it adds to your understanding.

--Handout:   "#2" (lefthanded families).  Make a tree and a table, with all probabilities.  (You should get P(L)=.092, P(Neither|L) = .529)
--Moore&McCabe 5th ed, p. 323 ff.  You did and kept  4.104, 105 using trees.  Now do 4.106, 107.
A. Hand this one in! (Monday if you need the time.  Work with anyone...) With Handout--the probabilities are now different--the tests have improved.  The issues are similar.  All positive screening results still should be confirmed with a much more accurate test (usually the "Western blot?").
   A test for HIV antibodies in mouth fluids (OraQuick) that gives results in half an hour is now used in quick screening contexts.  (http://aidsinfo.nih.gov/ContentFiles/FDA-3-3-06.pdf)  " The labeled sensitivity of the test is 99.3%... and the specificity is 99.8%...[Specificity may actually be lower in some circumstances:] 99.1%" .  Remember sensitivity is P(test pos|antibodies yes) and specificity is P(test negative |no antibodies).   Let's use 99.1% for the specificity.
a) Assume the proportion of the people with HIV antibodies in the population you're interested in is 1%.  Make a tree, and a table if you like.   What is the probability an individual chosen at random tests positive?  If an individual chosen at random tests positive, what is the probability (s)he actually has HIV antibodies?  If an individual chosen at random tests negative, what is the probability (s)he is actually HIV free?
b)  Repeat the analysis where the proportion in the population with HIV is x%.  Then answer the questions for x = 0.1% (say a blood donation drive population).  For x = 10% (say a (non-celebrity) drug rehab clinic population).

  Optional: M&M p. 327, 4.109-110 are very interesting. Read the problems for the flavor, at least.

--Ash sec. 2-1, p. 44
   Add to   3a from last time: 3b.
    4, 6

= = = = = = = = = = = = = = = = =
Bayes' type problems, continued.  (Recap Day 7)

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) :  Given proportions of murders who got the death penalty, and for those who did and didn't, what proportion murdered White victims.  Desire: P(Death sentence| Victim White) = P(D|W).   From handout tree we see 

P(W) = (30+184)/326.   P(D and W) = 30/326.  P(D|W) = 30/(30+184) = 30/214  = .14
We want to compare this with P(Death sentence| Victim not white) = P(D|B) = 6/(6+106) = 6/112 = .05  (Radelet's claim is roughly correct)