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14.1 The Basics of Probability
Theory
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 1
Sample Space and Events
Random phenomena are occurrences that vary
from day-to-day and case-to-case.
Although we never know exactly how a random
phenomenon will turn out, we can often calculate a
number called a probability that it will occur in a
certain way.
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 2
Sample Space and Events
• Example: Determine a sample space for the
experiment of selecting an iPhone from a
production line and determining whether it is
defective.
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Section 14.1, Slide 3
Sample Space and Events
• Example: Determine a sample space for the
experiment of selecting an iPhone from a
production line and determining whether it is
defective.
• Solution: This sample space is
{defective, nondefective}.
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Section 14.1, Slide 4
Sample Space and Events
• Example: Determine the sample space for the
experiment.
Three children are born to a family and
we note the birth order with respect to
gender.
(continued on next slide)
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Section 14.1, Slide 5
Sample Space and Events
• Solution: We use the tree diagram to find the
sample space:
{bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}.
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Section 14.1, Slide 6
Sample Space and Events
• Example: Determine the sample space for the
experiment.
We roll two dice and observe the pair of
numbers showing on the top faces.
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Section 14.1, Slide 7
Sample Space and Events
• Example: Determine the sample space for the
experiment.
We roll two dice and observe the pair of
numbers showing on the top faces.
• Solution: The
solution space
consists of the 36
listed pairs.
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Section 14.1, Slide 8
Sample Space and Events
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 9
Sample Space and Events
• Example: Write each event as a subset of the
sample space.
a) A head occurs when we flip a single coin.
b) Two girls and one boy are born to a family.
c) A sum of five occurs on a pair of dice.
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 10
Sample Space and Events
• Example: Write each event as a subset of the
sample space.
a) A head occurs when we flip a single coin.
b) Two girls and one boy are born to a family.
c) A sum of five occurs on a pair of dice.
• Solution:
a) The set {head} is the event.
b) The event is {bgg, gbg, ggb}.
c) The event is {(1, 4), (2, 3), (3, 2), (4, 1)}.
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Section 14.1, Slide 11
Sample Space and Events
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Section 14.1, Slide 12
Sample Space and Events
• Example: A pharmaceutical company is testing
a new drug. The company injected 100 patients
and obtained the information shown. Based on
the table, if a person is injected with this drug,
what is the probability that the patient will
develop severe side effects?
(continued on next slide)
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Section 14.1, Slide 13
Sample Space and Events
• Solution: We obtain the following probability
based on previous observations.
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Section 14.1, Slide 14
Sample Space and Events
• Example: The table summarizes the marital
status of men and women (in thousands) in the
United States in 2006. If we randomly pick a
male, what is the probability that he is divorced?
(continued on next slide)
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Section 14.1, Slide 15
Sample Space and Events
• Solution: We are only interested in males, so
we consider our sample space to be the
60,955 + 2,908 + 10,818 + 2,210 + 39,435 = 116,326
males.
The event, call it D, is the set of 10,818 men who
are divorced. Therefore, the probability that we
would select a divorced male is
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 16
Counting and Probability
Probabilities may based on empirical information.
For example, the result of experimental data.
Probabilities may based on theoretical
information, namely, combination formulas.
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Section 14.1, Slide 17
Counting and Probability
• Example: We flip three fair coins. What is the
probability of each outcome in this sample
space?
•
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Section 14.1, Slide 18
Counting and Probability
• Example: We flip three fair coins. What is the
probability of each outcome in this sample
space?
• Solution: Eight equally likely outcomes are
shown below. Each has a probability of
© 2010 Pearson Education, Inc. All rights reserved.
.
Section 14.1, Slide 19
Counting and Probability
• Example: We draw a 5-card hand randomly
from a standard 52-card deck. What is the
probability that we draw one particular hand?
•
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Section 14.1, Slide 20
Counting and Probability
• Example: We draw a 5-card hand randomly
from a standard 52-card deck. What is the
probability that we draw one particular hand?
• Solution: In Chapter 13, we found that there
are C(52, 5) = 2,598,960 different ways to
choose 5 cards from a deck of 52. Each hand
has the same chance of being drawn, so the
probability of any particular hand is
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Section 14.1, Slide 21
Counting and Probability
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Section 14.1, Slide 22
Counting and Probability
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Section 14.1, Slide 23
Counting and Probability
• Example: What is the probability in a family
with three children that two of the children are
girls?
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Section 14.1, Slide 24
Counting and Probability
• Example: What is the probability in a family
with three children that two of the children are
girls?
• Solution: We saw earlier that there are eight
outcomes in this sample space. We denote the
event that two of the children are girls by the set
G = {bgg, gbg, ggb}.
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Section 14.1, Slide 25
Counting and Probability
• Example: What is the probability that a total of
four shows when we roll two fair dice?
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Section 14.1, Slide 26
Counting and Probability
• Example: What is the probability that a total of
four shows when we roll two fair dice?
• Solution: The sample space for rolling two
dice has 36 ordered pairs of numbers. We will
represent the event “rolling a four” by F. Then
F = {(1, 3), (2, 2), (3, 1)}.
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Section 14.1, Slide 27
Counting and Probability
• Example: If we draw a 5-card hand from a
standard 52-card deck, what is the probability
that all 5 cards are hearts?
•
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Section 14.1, Slide 28
Counting and Probability
• Example: If we draw a 5-card hand from a
standard 52-card deck, what is the probability
that all 5 cards are hearts?
• Solution: We know that there are C(52, 5)
ways to select a 5-card hand from a 52-card
deck. If we want to draw only hearts, then we
are selecting 5 hearts from the 13 available,
which can be done in C(13, 5) ways.
© 2010 Pearson Education, Inc. All rights reserved.
Section 14.1, Slide 29
Counting and Probability
• Example: Four friends belong to a 10-member
club. Two members of the club will be chosen to
attend a conference. What is the probability that
two of the four friends will be selected?
•
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Section 14.1, Slide 30
Counting and Probability
• Example: Four friends belong to a 10-member
club. Two members of the club will be chosen to
attend a conference. What is the probability that
two of the four friends will be selected?
• Solution: We can choose 2 of the 10 members
in
Event E, choosing 2 of the 4 friends, can be done
in C(4, 2) = 6 ways.
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Section 14.1, Slide 31
Probability and Genetics
Y – produces yellow seeds (dominant gene)
g – produces green seeds (recessive gene)
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Section 14.1, Slide 33
Probability and Genetics
Crossing two first
generation plants:
Punnett Square
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Section 14.1, Slide 34
Probability and Genetics
• Example: Sickle-cell anemia is a serious
inherited disease. A person with two sickle-cell
genes will have the disease, but a person with
only one sickle-cell gene will be a carrier of the
disease. If two parents who are carriers of sicklecell anemia have a child, what is
the probability of each of the following:
a) The child has sickle-cell anemia?
b) The child is a carrier?
c) The child is disease free?
(continued on next slide)
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Section 14.1, Slide 35
Probability and Genetics
• Solution:
Use a Punnett square:
s denotes sickle cell
n denotes normal cell.
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Section 14.1, Slide 36
Odds
If a family has 3 children,
what are the odds against
all 3 children being of the
same gender? 6:2 or 3:1
What are the odds in
favor? 1:3
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Section 14.1, Slide 37
Odds
• Example: A roulette wheel has 38 equal-size
compartments. Thirty-six of the compartments
are numbered 1 to 36 with half of them colored
red and the other half black. The remaining 2
compartments are green and numbered 0 and
00. A small ball is placed on the spinning wheel
and when the wheel stops, the ball rests in one of
the compartments. What are the odds against the
ball landing on red?
(continued on next slide)
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Section 14.1, Slide 38
Odds
• Solution:
There are 38 equally likely outcomes. 18 are in
favor of the event “the ball lands on red” and 20
are against the event.
The odds against red are 20 to 18 or 20:18,
which we reduce to 10:9.
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Section 14.1, Slide 39
Odds
If the probability of E is 0.3, then the odds
against E are
We may write this as 70:30 or 7:3.
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Section 14.1, Slide 40
Odds
• Example: If the probability
of Green Bay winning the
Super Bowl is 0.35. What
are the odds against Green
Bay winning the Super
Bowl?
• Solution: From the diagram we compute
That is, the odds against are 13 to 7.
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Section 14.1, Slide 41