Download Slide 9- 1 Homework, Page 715

Homework, Page 715
Expand the binomial using a calculator to find the binomial
coefficients.
4
1.  a  b 
 4 4 0  4 3 1  4 2 2  4 3  4 0 4
 a  b    0  a b  1  a b   2  a b   3  ab   4  a b
 
 
 
 
 
4
 a 4  4a 3b1  6a 2b 2  4ab3  b 4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 1
Homework, Page 715
Expand the binomial using Pascal’s triangle to find the binomial
coefficients.
3
5.  x  y 
1
1
1
1
 x  y
3
1
2
3
1
3
1
 x3  3x 2 y  3xy 2  y 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 2
Homework, Page 715
Evaluate the expression by hand before checking your work.
9.  9 
 2
 
9 8 7!
9!
9
 2   2! 9  2 !  2 1 7!  9 4  36
 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 3
Homework, Page 715
Find the coefficient of the given
term in the binomial expansion.
14
11 3
x
y term in  x  y 
13.
14 
 3   14 C3  364
 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 4
Homework, Page 715
Use the Binomial Theorem to find the polynomial expansion of the
function.
5
f  x    x  2
17.
f  x    x  2
5
 5  5  5 4
5 3
 5 2
2
3
   x    x  2     x  2     x  2 
0
1 
 2
 3
5
 5
4
5
   x  2      2 
 4
 5
 x5  5 x 4  2   10 x3  4   10 x 2  8  5 x 16    32 
 x5  10 x 4  40 x3  80 x 2  80 x  32
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 5
Homework, Page 715
Use the Binomial Theorem to expand each expression.
4
21.  2x  y 
 4
 4
 4
 4
4
3
2 2
3  4 4
 2x  y    0   2 x   1   2 x  y   2   2 x  y   3   2 x  y   4  y
 
 
 
 
 
 16 x 4  4 8 x3 y  6 4 x 2 y 2  4  2 x  y 3  y 4
4

  
 
 16 x 4  32 x3 y  24 x 2 y 2  8 xy 3  y 4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 6
Homework, Page 715
Use the Binomial
Theorem to expand each expression.
5
2
x
3
25.




x  3   5  x 2 5   5  x 2 4  3   5  x 2 3  32   5  x 2 2  33
0
1 
 2
 3
 
 
 
 
 5  2
 5 5
4
   x  3     3
 4
 5
 x 10  5 x 8  3  10 x 6  9   10 x 4  27   5 x 2 81   243
2
5
 
 
 
 
 
 x 10  15 x 8  90 x 6  270 x 4  405 x 2  243
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 7
Homework, Page 715
n n 
 n for all integers n  1.
29. Prove that    

 1   n  1
n!
n
n  n  1!
n
1   1! n  1! 
1 n  1!
 
n  n  1!
n!
n 
 n  1  n  1! n   n  1 !   n  1!1!


n n 
   
n

 1   n  1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
n
Slide 9- 8
Homework, Page 715
 n   n  1
2
Prove
that


n
for all integers n  2.
33.
2 2 
  

n  n  1 n  2 ! n  n  1 n 2  n
n!
n


 2   2! n  2 ! 
2
2  n  2 !
2
 
 n  1!   n  1!   n  1 n  n  1!  n 2  n
 n  1 
 2  2!  n  1  2 ! 2  n  1!
2
2  n  1!


n 2  n n 2  n n 2  n  n 2  n 2n 2



 n2
2
2
2
2
 n   n  1
2
   

n

2
2
  

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 9
Homework, Page 715
37.
A.
B.
C.
D.
E.
What is the coefficient of x4 in the expansion of (2x + 1)8?
16
256
8 
4
4
4
1120

70
16x
2
x
1




 1120x 4
 4
 
1680
26,680
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Slide 9- 10
9.3
Probability
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about





Sample Spaces and Probability Functions
Determining Probabilities
Venn Diagrams and Tree Diagrams
Conditional Probability
Binomial Distributions
… and why
Everyone should know how mathematical the “laws of
chance” really are.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 12
Probability of an Event (Equally Likely
Outcomes)
If E is an event in a finite, nonempty sample space S of equally likely
outcomes, then the probability of the event E is
the number of outcomes in E
P( E ) 
.
the number of outcomes in S
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 13
Probability Distribution for the Sum of Two
Fair Dice
Outcome
2
3
4
5
6
7
8
9
10
11
12
Probability
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
1
2
3
4
5
6
1
2
3
4
5
6
7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
3
4
5
6
7
8
3 4 5 6
4 5 6 7
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 11
9 10 11 12
Slide 9- 14
Example Rolling the Dice
Find the probability of rolling a sum evenly divisible by 4 on a
single roll of two fair dice.
1
2
3
4
5
6
1
2
3
4
5
6
7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
3
4
5
6
7
8
3 4 5 6
4 5 6 7
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 11
9 10 11 12
Slide 9- 15
Probability Function
A probability function is a function P that assigns a real number
to each outcome in a sample space S subject to the following conditions:
1. 0  P(O)  1;
2. the sum of the probabilities of all outcomes in S is 1;
3. P()  0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 16
Example Testing the Validity of a
Probability Function
Is it possible to weight a standard number cube in such a way that
the probability of rolling a number n is exactly 1/(n+2)?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 17
Probability of an Event (Outcomes not
Equally Likely)
Let S be a finite, nonempty sample space in which every outcome
has a probability assigned to it by a probability function P. If E is
any event in S , the probability of the event E is the sum of the
probabilities of all the outcomes contained in E.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 18
Example Rolling the Dice
Find the probability of rolling a sum evenly divisible by 3 on a
single roll of two fair dice.
1
2
3
4
5
6
1
2
3
4
5
6
7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2
3
4
5
6
7
8
3 4 5 6
4 5 6 7
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 11
9 10 11 12
Slide 9- 19
Strategy for Determining Probabilities
1. Determine the sample space of all possible outcomes. When possible,
choose outcomes that are equally likely.
2. If the sample space has equally likely outcomes, the probability of an
the number of outcomes in E
event E is determined by counting: P ( E ) 
.
the number of outcomes in S
3. If the sample space does not have equally likely outcomes, determine
the probability function. (This is not always easy to do.) Check to be sure
that the conditions of a probability function are satisfied. Then the
probability of an event E is determined by adding up the probabilities
of all the outcomes contained in E.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 20
Example Choosing Chocolates
Dylan opens a box of a dozen chocolate crèmes and offers three of them to
Russell. Russell likes vanilla crèmes the best, but all the chocolates look alike
on the outside. If five of the twelve crèmes are vanilla, what is the probability
that all of Russell’s picks are vanilla?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 21
Multiplication Principle of Probability
Suppose an event A has probability p1 and an
event B has probability p2 under the assumption
that A occurs. Then the probability that both A
and B occur is p1p2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 22
Venn Diagram
Venn diagrams are visual representations of groupings of events.
For example, if 63% of the students are girls and 54% of the
students play sports, find the percentage of boys playing sports if
1/3 of the girls play sports.
Girls
Sports
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 23
Conditional Probability Formula
If the event B depends on the event A, then P( B | A) 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
P( A and B)
.
P( A)
Slide 9- 24
Example Using the Conditional
Probability Formula
Two identical cookie jars are on a counter. Jar A contains eight
cookies, six of which are oatmeal, and jar B contains four
cookies, two of which are oatmeal. If an oatmeal cookie is
selected, what is the likelihood it came from the jar A?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 25
Binomial Distribution
Suppose an experiment consists of n-independent repetitions of an
experiment with two outcomes, called "success" and "failure." Let
P (success)  p and P (failure)  q. (Note that q  1  p.)
Then the terms in the binomial expansion of ( p  q ) n give the respective
probabilities of exactly n, n  1,..., 2, 1, 0 successes.
Number of successes out of
Probability
n independent repetitions
n
pn
n 1
 n  n1
 n  1 p q


1
0
 n  n1
 r  pq
 
qn
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 26
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes all 15?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 27
Example Shooting Free Throws
Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws,
and if his chance of making each one is independent of the other shots, what is
the probability that he makes exactly 10?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 28
Homework



Homework Assignment #32
Read Section 9.4
Page 728, Exercises: 1 - 53 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 29
9.4
Sequences
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Evaluate each expression when a  3, r  2, n  4 and d  2.
1. a  (n  1)d
2. a  r
Find a .
n 1
10
k 1
k
4. a  2  3
3. a 
k
k 1
k
5. a  a  3 and a  10
k
k 1
9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 31
Quick Review Solutions
Evaluate each expression when a  3, r  2, n  4 and d  2.
1. a  (n  1)d 9
2. a  r
24
Find a .
n 1
10
k 1
k
4. a  2  3
3. a 
k
k 1
k
11
10
39,366
5. a  a  3 and a  10
k
k 1
9
13
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 32
What you’ll learn about




Infinite Sequences
Limits of Infinite Sequences
Arithmetic and Geometric Sequences
Sequences and Graphing Calculators
… and why
Infinite 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- 33
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- 34
Example of an Explicitly Defined
Sequence
Find the first 4 and the 50 th term of the sequence ak 
in which ak  k 2  3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 35
Example of a Recursively Defined
Sequence
Find the first 4 and the 50th term of the sequence ak 
in which a1  2, an  an 1  2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 36
Limit of a Sequence
Let an  be a sequence of real numbers, and consider lim an .
n 
If the limit is a finite number L, the sequence converges and L
is the limit of the sequence. If the limit is infinite or nonexistent,
the sequence diverges.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 37
Example Finding Limits of Sequences
Determine whether the sequence converges or diverges. If it converges,
give the limit.
2 1 2
2
2,1, , , ,..., ,...
3 2 5
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 38
Arithmetic Sequence
A sequence an  is an arithmetic sequence if it can be written in the
form a, a  d , a  2d ,..., a  (n  1)d ,... for some constant d .
The number d is called the common difference.
Each term in an arithmetic sequence can be obtained recursively from
its preceding term by adding d : an  an1  d (for all n  2).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 39
Example Arithmetic Sequences
Find (a) the common difference, (b) the tenth term, (c) a recursive rule for the
nth 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- 40
Geometric Sequence
A sequence an  is a geometric sequence if it can be written in the


form a, a  r , a  r 2 ,..., a  r n1 ,... for some nonzero constant r.
The number r is called the common ratio.
Each term in a geometric sequence can be obtained recursively from
its preceding term by multiplying by r : an  an1  r (for all n  2).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 41
Example Defining Geometric Sequences
Find (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- 42
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- 43
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- 44
Example Calculating Sequence Values
Use you calculator to generate the first 10 terms of the sequence
explicitly defined by a1 = 4, an = 3an-1 + 5 in a scatter plot.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 45
The Fibonacci Sequence
The Fibonacci sequences can be defined recursively by
a1  1
a2  1
an  an2  an1
for all positive integers n  3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 9- 46