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Chapter 6: Probability—
The Study of Randomness
6.1
The Idea of Probability
6.2
Probability Models
6.3
General Probability Rules
1
Simple Question:
• If tossing a coin, what is the probability of the
coin turning up heads?
• Most of you probably answered 50%, but how
do you know this to be so?
2
Probability
• Probability is the branch of mathematics that describes
the pattern of chance outcomes.
• The heart of statistics, and thus the heart of this course,
is in statistical inference.
– Probability calculations are the basis for inference.
• Mathematical probability is an idealization based on
imagining what would happen in an indefinitely long
series of trials.
3
Random Behavior
• Many observable phenomena are random: the
relative frequencies of outcomes seem to settle
down over the long haul.
• The big idea of probability: chance (“random”)
behavior is unpredictable in the short run, but
has regular and predictable patterns in the long
run.
• See Example 6.1, p. 331
4
Figure 6.1, p. 331
5
Simulation Problem
• Problem 6.3, p. 334
6
Probability Definitions
• Sample Space (S)—set of all possible outcomes.
– See Example 6.3, p. 336 (rolling two dice)
• Event—a particular outcome or set of
outcomes; a subset of the sample space.
• Probability—the number of times an event can
occur within the sample space divided by the
sample space.
– What is the probability of getting a sum ≤5
when rolling two dice?
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Figure 6.2, p. 336
8
Multiplication Principle
• Multiplication (counting) principle
– If you can do one task in a ways and a second task in b
ways, then both tasks can be done in a x b ways.
– How many outcomes are possible if we flip 4 coins?
– Apply the multiplication rule to the 2-dice problem.
• Example 6.5, p. 337
– Tree diagram
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Figure 6.3, p. 338
10
HW Exercises
• 6.11, p. 340
• 6.14 a, b p. 341
• 6.18, p. 342
• Reading, pp. 335-352
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Probability Rules (Box on p. 343)
• Rule 1. If P(A) is the probability of an event A,
then:
0  P( A)  1
• Rule 2. If S is the sample space in a probability
model, then:
P( S )  1
• Rule 3. The complement (Ac) of an event A is the
event that A does not occur. The complement
rule states:
P( A )  1  P( A)
c
12
Probability Rules, cont.
(Box on p. 343 and p. 351)
Rule 4. Two events are disjoint (mutually exclusive) if
they have no outcomes in common and can never occur
simultaneously. If A and B are disjoint, then:
P(A or B) = P(A) + P(B)
Rule 5. Two events A and B are independent if knowing
that one occurs does not change the probability that
the other occurs. If A and B are independent, then:
P( A and B)  P( A)  P( B)
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Independence Example
• What is the probability of drawing two hearts
on successive draws from a standard deck of
playing cards?
– With replacement
– Without replacement
14
Set Notation
A or B  A B... read, A union B.
A intersect B  A  B
For mutually exclusive events : A  B   (empty set)
15
In-Class Problems
• 6.19, p. 348
• Look at example 6.10, p. 345
– 6.26, p. 349
• More Problems:
• 6.27, 6.28, p. 354
• 6.31, 6.32, 6.33, p. 355
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HW
• Problems:
– 6.36, 6.37, p. 357
– 6.42, p. 359
• Read through p. 370.
17
More on Independence and Disjoint
(Mutually Exclusive) Events
• If I roll two dice, then each die can roll a 3. Neither die influences
the other, so they are independent. But since I can roll a 3 on each
die simultaneously, they are not disjoint events.
+++++++++++++++++++++++++++++++++++++++
• A bag contains 3 red balls and 2 green balls. A ball is drawn from
the bag, its color is noted, and the ball is set aside. Then a second
ball is drawn and its color is noted.
– Let event A be the event that the first ball is red. Let event B be
the event that the second ball is red.
• Events A and B are not disjoint because both balls can be red.
Events A and B are not independent because whether the first ball
is red or not alters the probability of the second ball being red.
+++++++++++++++++++++++++++++++++++++++
• Two events that are disjoint cannot be independent.
– See last full paragraph on p. 352.
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6.3 General Probability Models
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General Addition Rule for
Unions of Two Events
• Rule 4 (Addition Rule). Two events are disjoint
(mutually exclusive) if they have no outcomes in
common and can never occur simultaneously. If A and
B are disjoint, then:
P( A or B)  P( A)  P( B)
• The above rule does not work if two events are not
disjoint (that is, they are not mutually exclusive). Here
is the general rule for addition for unions of two events:
P( A or B)  P( A)  P( B)  P( A and B)
20
Venn Diagrams
• Venn Diagrams can be very useful in helping find
probabilities of unions.
21
Practice Problems
• 6.46, p. 364
• 6.47, p. 364
• 6.52, p. 365
• 6.51, p. 365
22
More Practice
• pp. 364-365: 6.48, 6.49, 6.50, 6.53
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Conditional Probability
• Probability of an event given, or under the
condition that, we know another event.
– See example 6.18, p. 366
• Notation:
P( A / B) or P( B / A)
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General Multiplication Rule
for any Two Events
P( A and B)  P( A)  P( B / A)
P( A and B)
P( B / A) 
P( A)
Note: If events A and B are independent, then: P ( B / A)  P ( B )
25
Practice Problems
• 6.54 and 6.55, pp. 369-370
26
HW
• Problems, pp. 370-371:
– 6.56, 6.58, 6.60
– 6.61 (include a Venn diagram for part b)
27
Tree Diagrams
• Can be used to solve a number of probability problems.
• Draw a tree diagram to represent this situation:
– 5% of male high school athletes go on to play college
sports. Of this group, 1.7% will go on to play
professionally.
– Of the group of high school athletes which does not
play college sports, only 0.01% will play
professionally.
• Question: What’s the probability that a high school
athlete will play professionally?
28
Tree Diagrams
• See Examples 6.22-6.23, pp. 372-373
• The probability of reaching the end of any
complete branch is the product of the
probabilities written on each segment.
– Probabilities after the first level of segments are
conditional probabilities.
29
Example 6.23, p. 372
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Example
• The weather forecaster at the radio station
reports that if it rains on a given day in
November, the probability of rain the next day
is 70%. If it does not rain on a given day, then
the probability of rain the next day is 40%.
What is the probability that it will rain on
Friday if it is raining on Wednesday?
– Draw a tree diagram to help solve this problem.
31
Practice Problems
• 6.70, p. 381
– Use a tree diagram to answer (b).
– Then, use the general multiplication rule to
confirm your answer.
• 6.62 p. 377
• 6.63, p. 378
• 6.64, p. 378
– Use a tree diagram.
• 6.67, 6.69 p. 380
• 6.73, 6.74 p. 381
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Additional Practice Problems
• Problems, pp. 383-385:
– 6.78, 6.79, 6.82, 6.83, 6.84, 6.86
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