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Probability
I Introduction to Probability
A Satisfactory outcomes vs. total outcomes
B Basic Properties
C Terminology
II Combinatory Probability
A The Addition Rule Or
1. The special addition rule (mutually exclusive events)
2. The general addition rule (non-mutually exclusive events)B The Multiplication Rule And
1. The special multiplication rule (for independent events)
2. The general multiplication rule (for non-independent events)
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Probability for Equally Likely Outcomes
Suppose an experiment has N possible outcomes, all equally
likely. Then the probability that a specified event occurs equals
the number of ways,f, that the event can occur, divided by the
total number of possible outcomes. In symbols
Probability of a given event =N
f
Number of ways a given event can occur
Total of all possible outcomes
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Frequency distribution of annual income for U.S.
families
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Probability from Frequency Distributions
What is the a priori probability
of having an income between
$15,000 and $24,999
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Frequency distribution for students ages
N= 40
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Frequency distribution for students ages
What is the likelihood of randomly selecting a student who is
older than 20 but less than 22?
What is the likelihood of selecting a student whos age is an
odd number?
What is the likelihood of selecting a student who is either 21
or 23?
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Sample space for rolling a die once
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Possible outcomes for rolling a pair of dice
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Probabilities of 2 throws of the die
What is the probability of a 1 and a 3?
What is the probability of two sixes? What is the probability of at least one 3?
2/36
1/3612/36
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The Sum of Two Die TossesSum Frequency
2 13 2
4 35 4
6 57 6
8 5
9 4
10 311 2
12 1
What is the probability that the
sum will be
5?
7?
What is the probability that the
sum will be 10 or more?
What is the probability that the
sum will be either 3 or less or 11
or more?
4/36
6/36
6/36
3/36 + 3/36
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Two computer simulations of tossing a balanced coin
100 times
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Basic Properties of Probabilities
Property 1: The probability of an event is always between 0
and 1, inclusive.
Property 2: The probability of an event that cannot occur is 0.
(An event that cannot occur is called an impossible event.)
Property 3: The probability of an event that must occur is 1.
(An event that must occur is called acertain event.)
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A deck of playing cards
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The event the king of hearts is selected
1/52
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The event a king is selected
1/13 = 4/52
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The event a heart is selected
1/4 = 13/52
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The event a face card is selected
3/13=13/52
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Sample Space and Events
Sample space: The collection of all possible
outcomes for an experiment.
Event: A collection of outcomes for the
experiment, that is, any subset of the sample
space.
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Probability Notation
IfEis an event, thenP(E) stands for the
probability that eventEoccurs. It is read theprobability of E
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Venn diagram for eventE
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Relationships Among Events
(notE): The event that Edoes not occur.
(A &B): The event that bothA andB occur.
(A orB): The event that eitherA orB or both
occur.
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Event (notE) whereEis the probability of drawing a
face card.
40/52=10/13
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An event and its complement
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The Complementation Rule
For any eventE,
P(E) = 1 P (~E).
In words, the probability that an event occurs equals 1
minus the probability that it does not occur.
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Combinations of Events
The Addition Rule Or
The special addition rule (mutually exclusive events)
The general addition rule (non-mutually exclusive events)
The Multiplication Rule And
The special multiplication rule (for independent events)
The general multiplication rule (for non-independent events)
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Venn diagrams for
(a) event (notE
)(b) event (A &B)
(c) event (A orB)
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Event (B & C)
1/13 X 1/4 = 1/52
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Event (B orC)
16/52 = 4/52 + 13/52-1/52
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Event (C&D)
3/52 = 3/13 X 1/4
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Mutually Exclusive Events
Two or more events are said to bemutually exclusive if at
most one of them can occur when the experiment is performed,
that is, if no two of them have outcomes in common
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Two mutually exclusive events
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(a) Two mutually exclusive events
(b) Two non-mutually exclusive events
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(a) Three mutually exclusive events (b) Three non-
mutually exclusive events (c) Three non-mutually
exclusive events
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The Special Addition Rule
If event A and event B are mutually exclusive, then
More generally, if events A, B, C, are mutually exclusive, then
That is, for mutually exclusive events, the probability that at least one of
the events occurs is equal to the sum of the individual probabilities.
( ) ( ) ( )BPAPBAP +=or
( ) ( ) ( ) ( ) ......oror CPBPAPCBAP ++=
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Non-mutually exclusive events
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The General Addition Rule
IfA andB are any two events, then
P(A orB) = P(A) + P(B) P(A &B).
In words, for any two events, the
probability that one or the other occurs
equals the sum of the individualprobabilities less the probability that both
occur.
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P(A or B): Spade or Face Card
P (spade) + P (face card) P (spade & face card) = 1/4 + 3/13 3/52
= 22/52
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The Special Multiplication Rule (for independent events)
If eventsA, B, C, . . . are independent, then
P(A &B & C & ) = P(A) P(B) P(C).
What is the probability of all of these events occurring:
1. Flip a coin and get a head
2. Draw a card and get an ace
3. Throw a die and get a 1
P(A &B & C) =P(A) P(B) P(C) = 1/2 X 1/13 X 1/6
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Conditional Probability: For non-independent events
The probability that eventB occurs given that eventA
has occurred is called aconditional probability. It is
denoted by the symbolP(B |A), which is read theprobability ofB givenA. We callA thegiven event.
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Contingency Table for Joint Probabilities
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Contingency table for age and rank of faculty members
(using frequencies)
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The Conditional-Probability Rule
IfA andB are any two events, then
In words, for any two events, the conditional
probability that one event occurs given that the other
event has occurred equals the joint probability of the
two events divided by the probability of the givenevent.
.
)(
)&()|(
AP
BAPABP =
)&( BAP
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.)(
)&()|(
AP
BAPABP =
The Conditional-
Probability Rule
P(R3 |A4 ) =
= 36/253
= 0.142
P(A4 |R3 ) =
= 36/320
= 0.112
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Joint probability
distribution (using
proportions)
.)(
)&()|(
AP
BAPABP =
P(R3 |A4 ) =
= 0.031/0.217
= 0.142
P(A4 |R3 ) =
= 0.031/.0275
= 0.112
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Contingency table of marital status and sex
(using proportions)
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.
)(
)&()|(
AP
BAPABP =Joint probability
distribution (using
proportions)
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The General Multiplication Rule
IfA andB are any two events, thenP(A &B) = P(A) P(B |A).
In words, for any two events, their joint probability
equals the probability that one of the events occurs times
the conditional probability of the other event given thatevent.
Note: Either
1) The events are independent and then
P(A &B) =P(A) P(B).
Or
2) The events are not independent and then a
contingency table must be used
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Independent Events
EventB is said to be independent of eventA if theoccurrence of eventA does not affect the probability that
eventB occurs. In symbols,
P(B |A) = P(B).
This means that knowing whether eventA has occurredprovides no probabilistic information about the
occurrence of eventB.
Class Fr So Ju Se
Male 40 50 50 40 | 180
Female 80 100 100 80 | 360
120 150 150 120 | 540
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Probability and the Normal Distribution
What is the probability of randomlyselecting an individual with an I.Q. between
95 and 115? Mean 100, S.D. 15.
Find thez-score for 95 and 115 andcompute the area between
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More Preview of Experimental DesignUsing probability to evaluate a treatment effect. Values that are extremely
unlikely to be obtained from the original population are viewed as
evidence of a treatment effect.
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