# STAT 2060 Lecture Notes - Lecture 3: Contingency Table, Mutual Exclusivity, Conditional Probability

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29 May 2017

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Department

Course

Professor

STAT*2060: Statistics for Business Decisions

Week 3 Lectures

1 Introduction to Probability

Before we get into how to calculate probabilities of various experiments (or procedures), we ﬁrst need

some terminology:

Probability Experiment: An experiment is a procedure (or process) through which we obtain some

outcome or measurement. In a probability experiment, this outcome is determined by some element of

chance.

Sample space (S): The set of all possible outcomes of an experiment.

EXAMPLES:

i) Rolling a 6-sided die:

ii) Flippling a coin once:

iii) Flipping a coin twice:

iv) Applying to a University:

v) Applying for a $450,000 mortgage:

Sample points: The individual outcomes that make up the sample space.

1

S={1, 2, 3, 4, 5, 6}

S={Heads,Tails}

S={Head-Head/ Head-Tail/ Tail-Tail/Tail-Head}

S={Accepted, Rejected, Waitlisted, Conditional Acceptance}

S={Approved, Rejected, Approved for lesser amount}

denote

Rolling the 6-sided die=> sample points are 1, 2, 3, 4, 5, 6.

An event is a subset of the sample space. In basic probability, we often denote events with a capital

letter (i.e. A, B, etc.) We can visualize events using a Venn Diagram:

i) In rolling a 6-sided die, we deﬁne event Ato be the event that we roll an even number.

ii) In ﬂipping a coin twice, we deﬁne event Ato be the event that we ﬂip two heads, and event Bto be

the event that we ﬂip one head and one tail

iii) In applying to a University, we deﬁne event Ato be the event that our application is NOT rejected.

If all of the sample points in S are equally likely, the the probability of an event is deﬁned as:

P(event) = number of points in the event

total number of points in the sample space

i) In rolling a fair 6-sided die, what is the probability of event A? That is, what is the probability that

we roll an even number?

ii) In ﬂipping a fair coin twice, what is the probability of event A(that we ﬂip two heads)? What is

the probability of event B(we ﬂip one head and one tail)?

iii) In applying to a University, what is the probability of event A, that our application is NOT rejected,

if we assume that all four decisions regarding our application are equally likely?

2

A= {2, 4, 6}

A={HH}

B={HT, TH}

A={Accepted, Conditional Acceptance, Waitlisted}

S

1

2

3

4

5

6

A

HH

TH

HT

A

B

Accepted

Conditional

Waitlisted

A

P(event)= 3/6

= 0.5

P(event)= 1/4

=0.25

P=2/4

=0.5

P=3/4

=0.75

TT

Not reasonable assumption

In the case where all the sample points in Sare not equally likely, we will need to use probability rules

and calculations to ﬁnd the probability of our event of interest.

1.1 Rules of Probability

There are two probability rules that apply to all events and sample spaces:

1. All probabilities of events lie between 0 and 1.

2. For any sample space, the total probability has to be 1.

Intersection of Events:

The intersection of events A and B is the event that both A and B occur at the same time. We denote

the intersection as A ∩B.

EXAMPLE:

i) In rolling a fair 6-sided die, deﬁne event Ato be that we roll an even number, and deﬁne event Bto

be that we roll a 2 or a 6. What is the intersection of Aand B?

In some cases, we can calculate the probability of the intersection using our earlier formula, where:

P(A∩B) = number of points in the intersection

total number of points in the sample space

In our 6-sided die example, the P(A∩B) is then:

ii) In ﬂipping a fair coin twice,deﬁne event Ato be that we ﬂip two heads, and deﬁne event Bto be

that we ﬂip at least one head.What is the intersection of Aand B? What is P(A∩B)?

In other cases, calculating P(A∩B) requires us to have further knowledge about the events in question,

and in particular whether or not the events are independent. We will come back to this idea shortly.

3

For any event the probability of that event, P(event) greater or equal to zero, P(event) less than equal to one

For any sample space S, P(S)=1 ie. one of outcomes must occur

A intersect B

A={2, 4, 6}

B={2, 6}

1

3

5

2

4

6

A

B

AnB={2,6}

P(AnB)=2/6

=0.33333

A={HH}

B=(HT, TH, HH}

AnB={H,H}

P(AnB)=1/4

=0.25