Coming Next Class: Review of Chapter One Homework Problems

AMS 311

Lecture 5

February 8, 2000

Coming next class: Review of Chapter one homework problems.

Last Classes:

Theorem 2.1 (Counting Principle)

If the set E contains n elements and the set F contains m elements, there are nm ways in which we can choose, first, an element of E and then an element of F.

Theorem 2.2 (Generalized Counting Principle)

Let E1, E2, , Ek be sets with n1, n2, , nk elements, respectively. Then there n1 n2 nk ways in which we can, first, choose an element of E1, then an element of E2, , and finally an element of Ek .

Theorem 2.3. A set with n elements has 2n subsets.

Theorem 2.4.

The number of distinguishable permutations of n objects of k types, where n1 are alike, n2 are alike, , nk are alike, and n=n1+n2++nk is

Theorem 2.5 (Binomial Expansion) For any integer n>0,

Theorem 2.7. (Stirling’s Formula)

where the sign  means

Chapter Three

Conditional Probability and Independence

Estimates of conditional probabilities are fundamental statistics in the health sciences. For example, age-specific mortality rates are an example of estimates of conditional probabilities.

Definition: If P(B)>0, the conditional probability of A given B, denoted by P(A|B), is

Example 3.2. From the set of all families with two children, a family is selected at random and found to have a girl. What is the probability that the other child of the family is a girl? Assume that in a two-child family all sex distributions are equally probable. Answer : 1/3.

Example 3.3. From the set of all families with two children, a child is selected at random and is found to be a girl. What is the probability that the second child of this girl’s family is also a girl? Assume that in a two-child family all sex distributions are equally probably.

Answer: ½.

Theorem 3.1.

Let S be the sample space of an experiment, and let B be an event of S with P(B)>0. Then

a)P(A|B)0, for any event A of S.

b)P(S|B)=1.

c)If A1, A2,  is a sequence of mutually exclusive events, then

Proof?

Reducing the sample space to get a conditional probability.

Example 3.8. Let’s make a deal!

On a TV game show, there are three curtains. Behind two of the curtains there is nothing, but behind the third there is a prize that the player might win. The probability that the prize is behind a given curtain is 1/3. The game begins with the contestant randomly guessing a curtain. The host of the show (master of ceremonies), who knows begind which curtain the prize is, will then pull back a curtain other than the one chosen by the player, and reveal the prize is not behind that curtain. The host will ot pull back the curtain selected by the player, nor will he pull back the one with the prize, if different from the player’s choice. At this point, the host gives the player the opportunity to change his choice of curtain and select the other one. The question is whether the player should change his choice. That is, has the probability of the prize being behind the curtain chosen changed from 1/3 to ½ or is it still 1/3? If it is still 1/3, the contestant should definitely change his choice. Otherwise, there is no point in doing so.