Download Lecture 7 - Stony Brook AMS

AMS 311, Lecture 7
February15, 2001
Administrative Announcements:
1. Two problem quiz next Thursday, one combinatorial problem and one
Bayes’ Theorem problem.
2. 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 [email protected].
3. 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 ( r  3 ) 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 P( A1 A2  An1 )  0, then
P( A1 A2 A3  An1 An )  P( A1 ) P( A2 | A1 ) P( A3 | A2 A1 ) P( An | A1 A2  An1 ).
Definition: Let {B1 , B2 ,, Bn } be a set of nonempty subsets of the sample space S of an
experiment. If the events B1 , B2 ,, Bn are mutually exclusive and
n

i 1
Bi  S , the set
{B1 , B2 ,, Bn } is called a partition of S.
Theorem 3.4. Generalized Law of Total Probabililty
If {B1 , B2 ,, Bn } is a partition of the sample space of an experiment and P(Bi)>0 for
i  1,2,  , n , then for any event A of S,
n
P( A)   P ( A | Bi ) P( Bi ).
i 1
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,
P( A| Bk ) P( Bk )
P( Bk | A) 
.
P( A| B1 ) P( B1 )  P( A| B2 ) P( B2 )   P( A| Bn ) P( Bn )
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
P( AB)  P( A) P( B).
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
X : S   R is called a random variable of the experiment if, for each interval
I  R, {s: X ( s)  I } is an event.
Definition: If X is a random variable, then the function F defined on (   ,  ) by
F (t )  P( X  t ) is called the distribution function of X.
I use the term cdf (cumulative distribution function) rather than distribution function.