Transcript Mathematical Ideas

Chapter 12
Probability
© 2008 Pearson Addison-Wesley.
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Chapter 12: Probability
12.1 Basic Concepts
12.2 Events Involving “Not” and “Or”
12.3 Conditional Probability; Events Involving
“And”
12.4 Binomial Probability
12.5 Expected Value
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Chapter 1
Section 12-1
Basic Concepts
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Basic Concepts
•
•
•
•
•
Historical Background
Probability
The Law of Large Numbers
Probability in Genetics
Odds
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Historical Background
Much of the early work in probability concerned
games and gambling. One of the first to apply
probability to matters other than gambling was
Pierre Simon de Laplace, who is often credited with
being the “father” of probability theory. In the
twentieth century a coherent mathematical theory of
probability was developed through people such as
Chebyshev, Markov, and Kolmogorov.
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Probability
The study of probability is concerned with random
phenomena. Even though we cannot be certain
whether a given result will occur, we often can
obtain a good measure of its likelihood, or
probability.
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Probability
In the study of probability, any observation, or
measurement, of a random phenomenon is an
experiment. The possible results of the experiment
are called outcomes, and the set of all possible
outcomes is called the sample space.
Usually we are interested in some particular
collection of the possible outcomes. Any such
subset of the sample space is called an event.
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Example: Tossing a Coin
If a single fair coin is tossed, find the probability that
it will land heads up.
Solution
The sample space S = {h, t}, and the event whose
probability we seek is E = {h}.
P(heads) = P(E) = 1/2.
Since no coin flipping was actually involved, the
desired probability was obtained theoretically.
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Theoretical Probability Formula
If all outcomes in a sample space S are equally
likely, and E is an event within that sample space,
then the theoretical probability of the event E is
given by
number of favorable outcomes n( E )
P( E ) 

.
total number of outcomes
n( S )
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Example: Flipping a Cup
A cup is flipped 100 times. It lands on its side 84
times, on its bottom 6 times, and on its top 10 times.
Find the probability that it will land on its top.
Solution
From the experiment it appears that
P(top) = 10/100 = 1/10.
This is an example of experimental, or empirical
probability.
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Empirical Probability Formula
If E is an event that may happen when an experiment is
performed, then the empirical probability of event E is
given by
number of times event E occurred
P( E ) 
.
number of times the experiment was performed
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Example: Card Hands
There are 2,598,960 possible hands in poker. If there
are 36 possible ways to have a straight flush, find the
probability of being dealt a straight flush.
Solution
36
P(straight flush) 
 .0000139
2,598,960
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Example: Gender of a Student
A school has 820 male students and 835 female
students. If a student from the school is selected at
random, what is the probability that the student
would be a female?
Solution
number of female students
P(female) 
total number of students
835

 .505
820 + 835
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The Law of Large Numbers
As an experiment is repeated more and more
times, the proportion of outcomes favorable to
any particular event will tend to come closer
and closer to the theoretical probability of that
event.
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Probability in Genetics
Gregor Mendel, an Austrian monk used the idea of
randomness to establish the study of genetics. To
study the flower color of certain pea plants he found
that: Pure red crossed with pure white produces red.
Mendel theorized that red is “dominant” (symbolized
by R), while white is recessive (symbolized by r).
The pure red parent carried only genes for red (R),
and the pure white parent carried only genes for
white (r).
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Probability in Genetics
Every offspring receives one gene from each parent
which leads to the tables below. Every second
generation is red because R is dominant.
1st to 2nd Generation
offspring
First R
Parent R
Second
Parent
r
r
Rr Rr
Rr Rr
2nd to 3rd Generation
offspring
First R
Parent r
Second
Parent
R
r
RR Rr
rR rr
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Example: Probability of Flower Color
Referring to the 2nd to 3rd generation table (previous
slide), determine the probability that a third
generation will be
a) red
b) white
Base the probability on the sample space of equally
likely outcomes: S = {RR, Rr, rR, rr}.
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Example: Probability of Flower Color
Solution
a) Since red dominates white, any combination with
R will be red. Three out of four have an R, so
P(red) = 3/4.
b) Only one combination rr has no gene for red, so
P(white) = 1/4.
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Odds
Odds compare the number of favorable
outcomes to the number of unfavorable
outcomes. Odds are commonly quoted in horse
racing, lotteries, and most other gambling
situations.
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Odds
If all outcomes in a sample space are equally
likely, a of them are favorable to the event E,
and the remaining b outcomes are
unfavorable to E, then the odds in favor of E
are a to b, and the odds against E are b to a.
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Example: Odds
200 tickets were sold for a drawing to win a new
television. If Matt purchased 10 of the tickets,
what are the odds in favor of Matt’s winning the
television?
Solution
Matt has 10 chances to win and 190 chances to
lose. The odds in favor of winning are 10 to 190,
or 1 to 19.
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Example: Converting Probability to Odds
Suppose the probability of rain today is .43. Give
this information in terms of odds.
Solution
43
We can say that P (rain)  .43 
.
100
43 out of 100 outcomes are favorable, so 100 – 43 = 57
are unfavorable. The odds in favor of rain are 43 to 57
and the odds against rain are 57 to 43.
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Example: Converting Odds to Probability
Your odds of completing College Algebra class are 16 to
9. What is the probability that you will complete the
class?
Solution
There are 16 favorable outcomes and 9 unfavorable.
This gives 25 possible outcomes. So
16
P (completion) 
 .64.
25
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