homework, page 728
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Homework, Page 728. A red die and a green die have been rolled. What is the probability of the event? 1.The sum is nine. Homework, Page 728. A red die and a green die have been rolled. What is the probability of the event? 5.Both numbers are even. Homework, Page 728. - PowerPoint PPT PresentationTRANSCRIPT
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1
Homework, Page 728A red die and a green die have been rolled. What is the probability of the event?
1. The sum is nine.
1
2
3
4
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8
1 2 3
9 10
5
6
4 5 6
11 12
49
36P
1
9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2
Homework, Page 728A red die and a green die have been rolled. What is the probability of the event?
5. Both numbers are even.
9Both even
36P 1
2
3
4
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8
1 2 3
9 10
5
6
4 5 6
11 12
1
4
or
1 1Both even
2 2P
1
4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3
Homework, Page 7289. Alrik’s gerbil cage has four compartments, A, B, C, and D. After careful observation, he estimates the proportion of time spent in each compartment and constructs the following table.
(a) Is this a valid probability function? Explain.No, this is not a valid probability function because the proportions total more than 1.(b) Is there a problem with Alrik’s reasoning? Explain.The table has the gerbil spending 110% of its time in the compartments. This cannot be, as it only has 100% of the time to apportion to the various compartments.
Compartment A B C D
Proportion 0.25 0.20 0.35 0.30
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4
Homework, Page 728Candy is produced in the following proportions:
13. A single candy is randomly selected from a newly-opened bag. What is the probability that the candy is red?
Red 0.2P
Color Brown Red Yellow Green Orange Tan
Proportion 0.3 0.2 0.2 0.1 0.1 0.1
1
5 or 20%
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5
Homework, Page 728A peanut candy is produced in the following proportions:
17. A single peanut candy is randomly selected from each of two newly-opened bags. What is the probability that both are brown?
Brown 0.3 0.3P
Color Brown Red Yellow Green Orange
Proportion 0.3 0.2 0.2 0.2 0.1
0.09 9%
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6
Homework, Page 728A peanut candy is produced in the following proportions:
21. A single peanut candy is randomly selected from each of two newly-opened bags. What is the probability that neither is yellow?
Not yellow 0.8 0.8P
Color Brown Red Yellow Green Orange
Proportion 0.3 0.2 0.2 0.2 0.1
0.64
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7
Homework, Page 728A card game uses a 24-card deck, containing 9 through ace of the usual four suits. Each hand has six cards. Find the probability.
25. A hand contains all four aces.
4 4 20 2
24 6
Four AcesC C
PC
0.0014115
3542
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8
Homework, Page 72829. If it rains tomorrow, the probability is 0.8 that John will practice his piano lesson. If it does not rain, there is only a 0.4 chance John will practice. Suppose the chance of rain tomorrow is 60%. What is the probability that John will practice tomorrow?
Practice Rain Practice|rainP P P No rain Practice|no rainP P
0.6 0.8 0.4 0.4 0.48 0.16 0.64
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9
Homework, Page 72833. Floppy Jalopy Rent-a-Car has 25 cars available for rental, 20 big bombs and five midsize cars. If two cars are selected at random, what is the probability that both are big bombs?
Big bombsP 20 2
25 2
C
C 190
300
19
30
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10
Homework, Page 72837. Explain why the following statement cannot be true. The probabilities that a computer salesperson will sell zero, one, two, or three computers in any one day are 0.12, 0.45, 0.38, and 0.15, respectively.
This statement cannot be true because the probabilities add to more than one.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 11
Homework, Page 72841. To complete the kinesiology requirement at Palpitation Tech, you must pass two classes chosen from aerobics, aquatics, defense arts, racket sports, recreational activities, rhythmic activities, soccer, gymnastics, and volleyball. If you decide to choose your two classes at random by drawing two class names from a box, what is the probability you will take racket sports and rhythmic activities?
racket & rhythmicP 2 2
9 2
C
C
1
36
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12
Homework, Page 728Ten dimes dated 1990 through 1999 are tossed. Find the probability.
45. Heads on all ten dimes
ten headsP10
1
2
1
1024
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Homework, Page 728Ten dimes dated 1990 through 1999 are tossed. Find the probability.
49. At least one head
at least one headP10
10
2 1
2
1023
1024
11
1024
1023
1024
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 14
Homework, Page 72853. The probability of rolling a five on a pair of fair dice is:
A.
B.
C.
D.
E.
1
4
1
5
1
6
1
91
11
1 2 3 4 5 6
1
2
3
4
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 1
5
6 0 11 12
45
36P
1
9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
9.4
Sequences
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16
What you’ll learn about
Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators
… and whyInfinite sequences, especially those with finite limits, are involved in some key concepts of calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17
Sequence
Sequence - an ordered progression of numbers Finite sequence - a sequence with a finite number of entries Infinite sequence - a sequence that continues without bound Explicitly defined sequence - a sequence for which any entry
may be written directly using the definition Recursively defined sequence - a sequence defined in such a
manner that one must know the prior entry before being able to write the next entry using the definition
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18
Example of an Explicitly Defined Sequence
th
2
Find the first 4 and the 50 term of the sequence
in which 3.
k
k
a
a k
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 19
Example of a Recursively Defined Sequence
th
1 1
Find the first 4 and the 50 term of the sequence
in which 2, 2.k
n n
a
a a a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20
Limit of a Sequence
Let be a sequence of real numbers, and consider lim .
If the limit is a finite number , the sequence and
is the . If the limit is infinite or nonexistent,
the se
n nn
a a
L L
converges
limit of the sequence
quence .diverges
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21
Example Finding Limits of Sequences
Determine whether the sequence converges or diverges. If it converges,
give the limit.
2 1 2 22,1, , , ,..., ,...
3 2 5 n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22
Arithmetic Sequence
A sequence is an if it can be written in the
form , , 2 ,..., ( 1) ,... for some constant .
The number is called the .
Each term in an arithmetic seque
na
a a d a d a n d d
d
arithmetic sequence
common difference
1
nce can be obtained recursively from
its preceding term by adding : (for all 2).n nd a a d n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23
Example Arithmetic SequencesFind (a) the common difference, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.-2, 1, 4, 7, …
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24
Geometric Sequence
2 1
A sequence is a if it can be written in the
form , , ,..., ,... for some nonzero constant .
The number is called the .
Each term in a geometric sequence
n
n
a
a a r a r a r r
r
geometric sequence
common ratio
1
can be obtained recursively from
its preceding term by multiplying by : (for all 2).n nr a a r n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25
Example Defining Geometric SequencesFind (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term.2, 6, 18,…
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26
Sequences and Graphing Calculators
One way to graph an explicitly defined sequence is as a scatter plot of the points of the form (k,ak).
A second way is to use the sequence mode on a graphing calculator.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27
Example Graphing a Sequential Scatter Plot
Use you calculator to generate the first 10 terms of the sequence explicitly defined by an = 3n - 5 in a scatter plot.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28
Example Calculating Sequence Values
Use you calculator to generate the first 10 terms of the sequence recursively defined by a1 = 4, an = 3an-1 + 5 in a scatter plot.
Lower case u in calculator entry is obtained by pressing 2nd and then 7.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 29
The Fibonacci Sequence
1
2
2 1
The Fibonacci sequences can be defined recursively by
1
1
for all positive integers 3.n n n
a
a
a a a
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 30
Homework
Homework Assignment #29 Review Section 9.4 Page 739, Exercises: 1 - 37 (EOO), 43, 45, 47 Quiz next time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
9.5
Series
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32
Quick Review
10
1
3
10
1
3
is an arithmetic sequence. Use the given information to find .
1. 5; 4
2. 5; 2
is a geometric sequence. Use the given information to find .
3. 5; 4
4. 5; 4
5. Find the sum of
n
n
a a
a d
a d
a a
a r
a r
2 the first 3 terms of the sequence .n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33
Quick Review Solutions
10
1
3
10
1
3
is an arithmetic sequence. Use the given information to find .
1. 5; 4
2. 5; 2
is a geometric sequence. Use the given information to find .
3.
41
19
1,310,7205; 4
4.
n
n
a a
a d
a d
a a
a r
a
2
5; 4
5. Find the sum of the first 3 terms of the sequence
81,920
14.
r
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 34
What you’ll learn about
Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series
… and why
Infinite series are at the heart of integral calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 35
Summation Notation
1 21
In , the sum of the terms of the sequence
, ,..., is denoted which is read "the sum of
from 1 to ." The variable is called the .
n
n k kk
a a a a a
k n k
summation notation
index of summation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 36
Sum of a Finite Arithmetic Sequence
1 2
1 21
1
1
Let , ,..., be a finite arithmetic sequence with common
difference . Then, the sum of the terms of the sequence is
...
2
2 ( 1)2
n
n
k nk
n
a a a
d
a a a a
a an
na n d
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 37
Example Summing the Terms of an Arithmetic Sequence
A corner section of a stadium has 6 seats along the front row. Each
successive row has 3 more seats than the row preceding it. If the top
row has 24 seats, how many seats are in the entire section?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 38
Sum of a Finite Geometric Sequence
1 2
1 21
1
Let , ,..., be a finite geometric sequence with common
ratio . Then the sum of the terms of the sequence is
...
1
1
n
n
k nk
n
a a a
r
a a a a
a r
r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 39
Example Summing the Terms of a Finite Geometric Sequence
11
Find the summation of the geometric sequence
1 1 12, , , , ,2 0.25
2 8 32
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 40
Partial Sums
Partial sums are the sums of a finite number of terms in an infinite sequence. In some instances, the partial sums approach a finite limit and the series is said to converge.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 41
Example Examining Partial Sums
For the following series, find the first five terms in the
sequence of partial sums. Which of the series appear
to converge?
a 0.1 0.01 0.001 0.0001 0.00001
b 10 15 22.5 33.75 50.625
c 1 1 1 1 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 42
Infinite Series
1 21
An infinite series is an expression of the form
... ...k nk
a a a a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 43
Sum of an Infinite Geometric Series
1
1The geometric series converges if and only
if | | 1. If it does converge, the sum is .1
k
ka r
ar
r
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 44
Example Summing Infinite Geometric Series
1
1
Determine whether the series converges. If it converges,
give the sum.
2 0.25k
k