Lecture 1

Sec 1.1-1.4

Outline

• Define terms with an example (outcome, sample space, event, an event space)

• Axioms of probability

Ficticious Experiment

• Experiment: flip a coin three times. Observe the sequence of heads and tails.

– Question: Assuming that the coin is fair, what is the maximum number of heads and tails?

Ficticious Experiment

• Experiment: flip a coin three times. Observe the sequence of heads and tails.

• Results:– hhh– hht– hth– htt– thh– tht– tth– ttt

outcome

sample space

Definition of an outcome: any possible observation of the experiment

Sample space: a set of all possible outcomes.

Another Example

Ficticious Experiment

• Procedure: flip a coin three times. Observe the sequence of heads and tails.

• Results:– hhh– hht– hth– htt– thh– tht– tth– ttt

An event is a set of outcomes with the first flip being a head.

Exercise

• What are the outcomes associated with each event?

Event A and event B

Event A and event B are mutually exclusive.

Event C and Event D

Event C

Event D

Event C and D are not mutually exclusive!

Exercise

• What are the probabilities for event A, B, C and D?

Notation

• In Set theory, a set is a collection of things.

x

B is a set.A is a subset of B, i.e. A

A subset can also be empty. A null set, an empty set has no element. The notation of a null is .

Intersection• The intersection of two sets C and D

is the set of all elements which are contained in both C and D.

Event C

Event D

𝐶∩𝐷

Union• The union of sets C and D is the set

of all elements that are either in C or in D, or in both.

C

Mutually Exclusive

𝐴∩𝐵=𝜙

Quiz

• Monitor three consecutive phone calls going through a telephone switching office. Class each one as a voice call (v) if someone is speaking, or a data (d) if the call is carrying a modem or fax signal. Your observation is a sequence of three letters each letter is either v or d.

• Write down the following events:– First call is a voice call– One or more voice call

Answer

• Write down the following events:– First call is a voice call

A1={vvv,vvd,vdv,vdd}

– One or more voice callA4={vvv,vvd,vdv,dvv,vdd,dvd,ddv}

Telephone Usage

What is the sample space?What is the probability of the sample space?

Telephone Usage

What is the probability associated with the long call?

Telephone Usage

What is the probability associated with the brief callif you know the probability of a long call?