AMS578
AMS 311
Feb 15, 2000
Exam 1: Feb 24; Chapters One, Two, and Three.
My exam week office hours:
Friday (Feb 18): 1-2:30; Monday (Feb 21): 1-2:30; Tuesday (Feb22): 11:15-1:30; Wednesday (Feb23) 10-12.
Chapter One Problems
Page 10, 14. Prove De Morgan’s second law,
a) by elementwise proof;
b) by applying DeMorgan’s first law to Ac and Bc.
Page 25, 14:
Let A, B, and C be three events. Prove that
Review of last class
Law of Total Probability: With conditions given in text,
One important application of the law is to the calculation of biostatistical rates of death or disease incidence. The following definitions are used:
Crude rate in a population:
Directly standardized rate DS(D):
The probability of being in the i-th group in the coverage has been changed from the population value P(Bi) to RP(Bi), the corresponding value of an accepted reference group. In applications, this is often the United States population figure for the latest census.
Gambler’s Ruin Problem
Two gamblers play the game of “heads or tails,” in which each time a fair coin lands heads up, player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that player A initially has a dollars and player B has b dollars. If they continue to play this game successively, what is the probability that (a) A will be ruined; (b) the game goes forever with nobody winning?
Bayes Formula
Theorem 3.6 (Bayes’ Theorem)
Let {B1, B2, (, Bn} be a partition of the sample space S of an experiment. If for i=1, 2, (, n, P(Bi)>0, then for any event A of S with P(A)>0,
Example 3.19
A box contains seven red and 13 blue balls. Two balls are selected at random and are discarded without their colors being seen. If a third ball is drawn randomly and observed to be red, what is the probability that both of the discarded balls were blue?
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