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381
Probability-I
(Introduction to Probability)
QSCI 381 – Lecture 7
(Larson and Farber, Sect 3.1+3.2)
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Probability is to Statistics like a
Bat is to Baseball!
Probability Experiments
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
A
is an action or
trial, through which specific results (counts,
measurements, or responses) are obtained. The
result of a single trial in a probability experiment is
an
. The set of all possible outcomes of
a probability experiment is the
.
An
consists of one or more outcomes and is a
subset of the sample space.
Probability
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
Probability
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Implies chance and uncertainty
How do we measure it?
How do probabilities behave?
Probability Experiments
(Examples-I)
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Probability Experiment
Count of the number of
trees in a stand
Sample Space
{0, 1, 2, ….}
Event
>10
Outcome
5
Probability Experiments
(Examples-II)
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Probability Experiment
Sample Space
Determine the maturity
state of an baleen
whale
Mature, Calf, Juvenile
Event
Immature = {Juvenile
OR Calf}
Outcome
Calf
Probability Experiments
(Simple Events)
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

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An event that consists of only one outcome is
called a
(or an elementary
event)
Obtaining a maturity state of “mature” is a
simple event.
An event of a count of 5 or less trees in a
stand is not a simple event because it
consists of 6 possible outcomes:
{0, 1, 2, 3, 4, 5}.
Classical Probability-I
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
The
(or theoretical) probability
is used when each outcome in the sample
space is equally likely to occur. The classical
probability for an event E is given by:
P( E ) 
Number of outcomes in E
Total number of outcomes in the sample space
The probability of the event E.
Classical Probability-II
(Example)
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

What is the (classical) probability of
selecting the King of Clubs from a pack
of 52 cards?
If you drew a card from a pack and it
was the King of Clubs, what is the
probability of then randomly drawing
another club?
Empirical Probability
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
(or statistical) probability is
based on observations obtained from
probability experiments. The empirical
probability of event E is the relative
frequency of event E, i.e.
Frequency of event E f
P( E ) 

Total frequency
n
Empirical Probability
(Examples)
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
A lake consists of male and female fish. Males
and females are equally susceptible to
capture. You sample (and then release) 60
animals, of which 40 are female.
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What event are we interested in?
What is the probability of the event?
What is the probability of catching a female the
next time you fish?
How important are the assumptions about equal
susceptibility and releasing fish?
The Law of Large Numbers
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
As an experiment is repeated over and
over, the empirical probability of an
event approaches the theoretical
(actual) probability of the event
Subjective Probability
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


probabilities arise from
intuition, educated guesses and
estimates.
The probability that the Seahawks will
win the Super Bowl next year is ….
It is sometimes not that easy to
distinguish between subjective and
empirical probabilities.
Probabilities Formalized
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
A probability cannot be negative and cannot exceed
1, i.e.:
0  P( E )  1

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The probability of an event in the sample space (the
set of all possible outcomes) is 1.
The complement of the event E is the set of
outcomes not part of E. The probability of the
complement of E (denoted E’ , E-prime) is:
P ( E ')  1  P( E )
Examples
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
40 fish are sampled from a lake with a large
population. If 29 are female, what is the probability
of sampling a male?
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Probability of sampling a female, P(E) = ?
Probability of sampling a male is 1-P(E) = ?
If the lake only had 80 fish and the sex ratio was
50:50 initially, what is the probability of the 41st fish
being a male?
Genetics: Two snapdragons (Red and White) are
crossed, the possible outcomes are RR, RW, WR and
WW. What is the probability of the events: Red, Pink
(=white+red), and White?
Example (from a Test)
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The sex ratio at birth for a particular pinniped is
55:45 male:female. You select two pups at random
from the population. What is the probability that:
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a. they are both male?
b. at least one of them is male?
Solution:
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We can assume that the population is “large” (not 100!)
P(male) * P(male) = 0.55*0.55
P(at least one male) = 1-P(both female) = 1 - 0.45*0.45
Conditional Probability-I
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
A
is the
probability of an event occurring given
that another event has already
occurred. The conditional probability of
event B occurring given that event A
has already occurred is denoted P(B|A)
and is read as “probability of B given A”
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Conditional Probability-II
Gene
Present
Gene
Not Present
Total
High IQ
33
19
52
Normal IQ
39
11
50
Total
72
30
102
• What is the probability of having a high IQ?
• What is the probability of having a high IQ if you have the gene?
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Conditional Probability-III
Gene
Present
Gene
Not Present
Total
High IQ
33
19
52
Normal IQ
39
11
50
Total
72
30
102
• What is the probability of having a high IQ?
• Solution: This question says nothing about the other factor (gene)
so we look at the last column (total); =52/102
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Conditional Probability-IV
Gene
Present
Gene
Not Present
Total
High IQ
33
19
52
Normal IQ
39
11
50
Total
72
30
102
•What is the probability of having a high IQ if you have the gene?
• Solution: The question relates only to having the gene so we
are dealing with a conditional probability. We focus on the “Gene
present” column and find the probability of having a High IQ GIVEN
having the gene = 33/72.