AMS 311, Lecture 7
February15, 2001
Administrative Announcements:
- Two problem quiz next Thursday, one combinatorial problem and one Bayes’ Theorem problem.
- Our TA is Mr. Taewon Lee. His office hours are W and F from 3 to 4 pm in Math Tower: 3-129A. His e-mail address is .
- Chapter Four homework (due Feb 22): Starting on page 140: 2, 9*, 10, 14; starting on page 148: 4, 7*, 12; starting on page 159: 2, 6, 8*, 16; starting on page 168: 4, 8.
Chapter Two Problems
Section 2.2: 30. What is the probability that a random r digit number () contains at least one 0, at least one 1, and at least one 2?
Example problem (also related to Fisher’s Tea-tasting Lady)
The Great Carsoni, a magician, claims to have extrasensory perception. In order to test this claim, he is asked to identify the 4 red cards our of 4 red and 4 black cards which are laid face down on the table. The Great Carsoni correctly identifies 3 of the red cards and incorrectly identifies 1 of the black cards. Therefore, he claims to have proved his point. What is the probability that the Great Carsoni would have correctly identified at least 3 of the red cards if he were, in fact, guessing? (Regard the 4 cards selected by the Great Carsoni as an unordered sample of size 4).
Review of last class:
The following is always true:
Theorem 3.2. (Generalization of the Law of Multiplication):
If then
Definition: Let be a set of nonempty subsets of the sample space S of an experiment. If the events are mutually exclusive and the set is called a partition of S.
Theorem 3.4. Generalized Law of Total Probabililty
If is a partition of the sample space of an experiment and P(Bi)>0 for , then for any event A of S,
Bayes’ Theorem
Let be a partition of the sample space S of an experiment. If for then for any event A of S with P(A)>0,
Example problem:
Diseases D1, D2, and D3 cause symptom A with probabilities 0.5, 0.7, and 0.8, respectively. If 5% of a population have disease D1, 2% of a population have disease D2, and 3.5% of a population have disease D3, what percent of the population have symptom A? Assume that the only possible causes of symptom A are D1, D2, and D3 and that no one carries more than one of these three diseases.
Definition: Two events A and B are called independent if
If two events are not independent, they are called dependent. If A and B are independent, we say that {A, B} is an independent set of events.
Reasons for events to be independent:
- Device has been constructed to have independent outcomes (roulette wheels, etc.).
- A sample has been taken following the precise rules.
- Experimental units have been randomly assigned to treatments.
To show two events are independent, apply the definition.
Chapter Four: Distribution Functions and Discrete Random Variables
Definition: Let S be the sample space of an experiment. A real-valued function is called a random variable of the experiment if, for each interval is an event.
Definition: If X is a random variable, then the function F defined on by is called the distribution function of X.
I use the term cdf (cumulative distribution function) rather than distribution function.