Lecture 2

Zhihua (Sophia) Su

University of Florida

Jan 9, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Some Examples to Understand the Formal Definition ofProbabilityFundamental Principle of CountingEvaluating Probabilities Using Permutations

Reading assignment: Chapter 2: 2.4, 2.5

STA 4321/5325 Introduction to Probability 2

Some Examples to Understand the FormalDefinition of Probability

Random experiment: Choose a person from 4 persons atrandom, with no preference to any person.

What is the sample space?

What is the set of all possible events?

How to provide a probability assignment P , which reflectsour belief in how the random experiment is conducted, andalso satisfies the 3 axioms?

STA 4321/5325 Introduction to Probability 3

Some Examples to Understand the FormalDefinition of Probability

Random experiment: There are 3 mailboxes. Three people comeand put letters at random (with no preference to any of thethree mailboxes).

Task: Compute the probability of the event that each mailbox ischosen once.

STA 4321/5325 Introduction to Probability 4

Fundamental Principle of Counting

Do we always need to go through this procedure for calculatingprobabilities of events?

No. We can often use counting rules to get around the situation.

Counting rule 1

Suppose we are performing an experiment where all outcomesare equally likely, hence we assign the same probability to everysample point (say there are N sample points).Suppose the event of interest, say A, consists of nA samplepoints. Then,

P (A) =nA

N=

# sample points in A# total sample points

.

STA 4321/5325 Introduction to Probability 5

Fundamental Principle of Counting

Fundamental Principle of Counting

Suppose that an experiment consists of two successive tasks.The first task can result in n1 outcomes and for each outcome,the second task can result in n2 outcomes. Then the totalnumber of outcomes of the experiment is n1n2.

STA 4321/5325 Introduction to Probability 6

Fundamental Principle of Counting

Example: Birthday problem.

Random experiment: Choose 25 people.

Task: Evaluate the probability of the event that there is at least1 match in the 25 birthdays.

STA 4321/5325 Introduction to Probability 7

Evaluating Probabilities Using PermutationsPermutations:

Here are some identities which we have been using informallyuntil now, and which are quite helpful in counting principles fordetermining probabilities of events.

What is the number of ways of choosing r objects withreplacement from n objects (order is important)?

What is the number of ways of choosing r objects withoutreplacement from n objects (order is important)?

STA 4321/5325 Introduction to Probability 8