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

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29 May 2017
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Course
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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.
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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={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
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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(AB) = number of points in the intersection
total number of points in the sample space
In our 6-sided die example, the P(AB) 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(AB)?
In other cases, calculating P(AB) 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
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