Probability

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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
• The special addition rule (mutually exclusive
  events)
• The general addition rule (non-mutually exclusive
  events)
• B The Multiplication Rule And
• The special multiplication rule (for independent
  events)
• 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
Number of ways a given event can occur
Probability of a given event
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
1/36
12/36
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The Sum of Two Die Tosses
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 a certain 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
3/1313/52
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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
If E is an event, then P(E) stands for the
probability that event E occurs. It is read the
probability of E
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Venn diagram for event E
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Relationships Among Events
(not E) The event that E does not occur. (A
B) The event that both A and B occur. (A or
B) The event that either A or B or both
occur.
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Event (not E) where E is the probability of
drawing a face card.
40/5210/13
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An event and its complement
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The Complementation Rule
For any event E, 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 (not E )(b) event
(A B) (c) event (A or B)
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Event (B C)
1/13 X 1/4 1/52
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Event (B or C)
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 be mutually
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.
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Non-mutually exclusive events
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The General Addition Rule
If A and B are any two events, then P(A or B)
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 individual
probabilities 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 events A, B, C, . . . are independent, then
• P(A B C ¼) P(A) P(B) P(C)¼.
• What is the probability of all of these events
  occurring
• Flip a coin and get a head
• Draw a card and get an ace
• 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 event B occurs given that
event A has occurred is called a conditional
probability. It is denoted by the symbol P(B
A), which is read the probability of B given A.
We call A the given 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
If A and B 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
given event.
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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)
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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Joint probability distribution (using proportions)
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The General Multiplication Rule
If A and B are any two events, then P(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
that event.
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
Event B is said to be independent of event A if
the occurrence of event A does not affect the
probability that event B occurs. In symbols, P(B
A) P(B). This means that knowing whether
event A has occurred provides no probabilistic
information about the occurrence of event B.
Class Fr So Ju Se Male 40 50 50 40
180 Female 80 100 100 80 360 120 150 150 12
0 540
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Probability and the Normal Distribution

• What is the probability of randomly selecting an
  individual with an I.Q. between 95 and 115? Mean
  100, S.D. 15.
• Find the z-score for 95 and 115 and compute the
  area between

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More Preview of Experimental Design Using
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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A Preview of Sampling Distributions

• What is the probability of randomly selecting a
  sample of three individuals, all of whom have an
  I.Q. of 135 or more?
• Find the z-score of 135, compute the tail region
  and raise it to the 3rd power.
• This concept is critical to understanding future
  concepts

So while the odds chance selection of a single
person this far above the mean is not all that
unlikely, the odds of a sample this far above the
mean are astronomical
z 2.19
P 0.0143
0.01433 0.0000029
X
X
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Summary

• For multiple events there are two rules
• AND (multiplication) and OR (addition)
• There are just a few special considerations
• For the And rule, if the events are not
  independent, you dont multiply, you use a table.
• For the Or rule, if the events are not mutually
  exclusive you have to subtract off their double
  count