7B
Day
Chapter 7B: Probability
M328, Advanced Algebra 2
Topic and In-class Work
7B
Homework Assignment
Turned
in?
Thursday,
May 9
Friday,
May 10
Monday,
May 13
Tuesday,
May 14
Wednesday,
May 15
Thursday,
May 16
Friday,
May 17
Monday,
May 20
Tuesday,
May 21
Wednesday,
May 22
Thursday,
May 23
Friday, May
24 (1/2 Day)
Tuesday,
May 28
Wednesday,
May 29
Thursday,
May 30
Section 7.4- Day 1: Binomial Theorem
 Notes (p3-6)
Section 7.4- Day 2: Binomial Theorem
 Notes (p7-8)
**Graded Homework #13 Due**
Section 7.6- Day 1: Counting
 Notes (p9-12)
Section 7.6- Day 2: Permutations and
Combinations
 Notes (p17-21)
Quiz Review: 7.4 and 7.6
 Notes (p25-26)
 Worksheet 7.6C (p27-29)
**Quiz 7.4-7.6**

Page 690 #1-15 odd, 21, 23, 25, 27, 31,
33

Page 691 #35, 41-46

Worksheet 7.6A- Counting (p13-16)

Worksheet 7.6B- Permutations and
Combinations (p22-24)

Worksheet Quiz Review 7.4 and 7.6
(p30-31)
Section 7.7- Day 1: Odds, Complements,
Sample Space
 Notes (p32-34)
**Graded Homework #14 Due**
Section 7.7- Day 2: Independent,
Dependent, Binomial
 Notes (p35-37)
Section 7.7- Day 3:Mutually Exclusive
and Inclusive
 Notes (p41-42)
Quiz Review 7.7
 Notes (p45)
**Quiz 7.7**

Page 716 #1-6, 7, 12, 13, 14, 15, 17

Worksheet 7.7A- Independent vs
Dependent (p38-40)

Worksheet 7.7B- Mutually Exclusive
and Inclusive (p43-44)

Worksheet Quiz Review 7.7 (p46-47)

Worksheet- Conditional Probability
(p49-52)


Worksheet- Venn Diagram (p53-59)
Test Review #1 (p60-62)


Test Review #2 (p64-66)
Page 726 #43, 44, 47, 49, 55-59, 62-70
(skip 68), 73, 74
 Warmup- Review (p48)
**Graded Homework #15 Due**
Conditional Probability
 Notes (p49-52)
Venn Diagram
 Worksheet- Venn Diagram (p53-59)
Test Review
 Warmup- Review (p63)
1
Friday,
May 31
Monday,
June 3
Tuesday,
June 4
**Probability Test**
**All Graded Homeworks Due**
Review for Final Exams
Review for Final Exams
Final Exams
Chapter 7 Even Answers:
Section 7.1
2. 3, 9, 15, 21, 27
4.
3 4 15 12
10. 0, 5 , 5 , 17 , 13
6 5 4 3
, , , ,2
12. 7, 7, 9, 13, 19
consisting of natural numbers
24. -1, -5, -9, -13
Section 7.2
70. 429
6. -2, 8, -24, 64, -160
5 4 3 2
8. 2, -3, 4, -5, 6
14. A sequence is a function with a domain
16. Finite
18. Finite
20. Infinite
22. Infinite
26. 2, 5, 7, 12
72. 465
74. 170m
Section 7.3
50. yes; the common ratio is b.
62. $26,214.40
82. The common ratio is 2
Review Pg. 683
26. .9 + .09 + .009 + …; The sum is 1.
Review Pg. 725
2. -2, 4, -8, 16, -32; geometric
8. 12, 10, 8, 6, 4
4. 2, 6, 12, 20, 30; neither
10. 6, 12, 24, 48, 96
16. 222
18. -162
20. 45
28. 248
30. Does not exist
38. 2.7
40.
12. -11; -13 + 2n
22.
3
9
13
24. 70
36
27
−9
32. 2 + 8 + 32 +. ..
34. 20
20
 4i  6
i 1
42.
12
6. 1, 3, 4, 7, 11; neither
i
 i 1
i 5
2
14. 20
26. 125
36. 1
M328
Chapter 7- Probability
Section 7.4- Day 1- Binomial Theorem
Name: _________________
Date: __________________
Expand:
(x + y) 0 =
(x + y)1 =
(x + y)2 =
(x + y)3 =
(x + y)4 =
(x + y)5 =
Would you want to find (x + y)10?
There is a pattern for expanding a binomial. What do you notice?
How many terms in the expansion?
What do you notice about the variables?
Let’s look at the coefficients…
3
For reference, I have included row 0 to 14 of Pascal's Triangle
1
1
1
1
1
1
1
1
1
1
1
1
1
1
14
10
36
20
35
84
15
35
56
1
6
21
70
126
1
5
56
126
1
7
28
84
1
8
36
1
1
120
210
252
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
12
66
220
495
792
924
792
495
220
66
12
1
13
78
286
715
1287 1716 1716 1287 715
286
78
13
1
91
364
1001 2002 3003 3432 3003 2002 1001
364
91
14
1
9
4
10
15
28
1
6
10
21
1
3
4
6
8
3
5
7
2
9
45
1
Example using Pascal’s Triangle to find the coefficients.
(x + y)6 =
A more efficient way is using factorial notation….
5!
8!
0!
Let’s go back and look at the coefficients of the binomial expansion. Can we get the coefficient
by using the exponent of the term?
1st term:
2nd term:
3rd term:
4
4th term:
5th term:
6th term:
7th term:
Can we generalize and make a formula for find the coefficient?
Example:
3
3
C0   
0
Example:
6
6
C2   
 2
Generalize how to expand a binomial:
(x + y) n =
5
Expand
(x + y)8 =
(a - b)7 =
(m+ n2)4 =
(4a – 5b)5 =
6
M328
Chapter 7- Probability
Section 7.4- Day 2- Binomial Theorem
Name: _________________
Date: __________________
Warmup
1. Write the formula for :
n
n Cr   
r 
2. Expand (3r – y)6 =
4
m

3. Expand   n  =
4

7
M328
Chapter 7- Probability
Section 7.4- Day 2- Binomial Theorem
Name: _________________
Date: __________________
Would you want to expand the entire binomial just to find one term?
Expand (x + y)3
1st term
2nd term
3rd term
4th term
What do you notice?
Write a formula for finding a specific term of a binomial expansion.
Example: Find the 7th term of (a + 2b)10
Example: Find the 4th term of (2c – d)12
8
M328
Chapter 7- Probability
Counting
Name: _________________
Date: __________________
Tree Diagram: A restaurant offers 3 salads (house, spinach, chop) and 2 dressing types (ranch
and Italian). Make a tree diagram to show all possible salad and dressing outcomes.
In this situation, each salad represents an event. Two events are independent events if neither
influences the outcome of the other.
Fundamental Principle of Counting:
If n independent events occur, with
m1 ways for event 1 to occur
m2 ways for event 2 to occur
.
.
.
mn ways for event n to occur.
Then there are m1 ∙ m2 ∙ ∙ ∙mn different ways for all n events to occur.
Example 1: Jim has 4 pairs of pants, 3 shirts and 2 jackets. How many different outfits can he
wear?
9
Example 2: A teacher has 5 different books that she wishes to arrange in a row on a shelf. How
many different arrangements are possible?
n-factorial : 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)…(3)(2)(1)
Example 3: 6! =
and
Example 4:
0! = 1
9!
7!
=
Example 5: How many ways can the letters of the given word be arranged?
a) EXPENSIVE
b) MATHEMATICS
Example 6: How many four letter words can be made using the letters of the alphabet:
a) with repetition
b) without repetition
Example 7: In how many ways can 5 people line up for service in a cafeteria if 2 of them insist
on standing together?
10
Example 8: List all possible 3 digit numbers that can be formed from 1,3,5,7 if repetition is
not allowed.
Example 9: How many possible 2 digit numbers can be formed from 0,1, 2,3, 4 without
repetition?
Example 10: How many possible 2 digit numbers can be formed from 0,1, 2,3, 4 with
repetition?
Example 11: How many possible odd 4 digit numbers can be formed from 0,1, 2,3, 4 without
repetition?
11
Cards:
________ cards
________ suits
________ black spades
________ black clubs
________ red hearts
________ red diamonds
________ face cards
How many 2 card hands are possible?
How many 5 card hands are possible if all 5 cards are of the same suit?
12
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet 7.6A- Counting
Name: _______________________
1. List the possible ways can a consonant and then a vowel be chosen from the letters c, o, a, s,
t? Draw a tree diagram to represent the possible choices.
2. If a cafeteria offers 3 choices of appetizer, 4 main courses, 3 desserts and 3 beverages, how
many different complete meals are available?
3. A combination lock has 4 dials with 10 digits on each dial. How many combinations are
possible?
4. How many odd numbers of 3 digits can be written using the digits 1, 2, 3, 4 without
repetition? Draw a tree graph to list the possible choices.
5. How many 3 digit numbers can be written using the digits 0, 1, 2, 3 without repetition?
13
6. How many 3 letter “words” can be made using the letters a, b, c, d, e
a) without repetition?
b) repetition is allowed?
7. How many 3 letter “words” can be made using the letters of the alphabet:
a) without repetition?
b) repetition is allowed?
8. In how many ways can 3 different letters be mailed if there are 5 mailboxes?
9. A family of six has two rows of three seats each in an airplane. In how many ways can they
be seated so that the two youngest children have the two window seats?
10. How many 5 digit numbers with repetition are there in which the sum of the first and last
digits is 6?
14
11. a) In how many ways can a consonant and a vowel be selected from the alphabet of 21
consonants and 5 vowels?
b) Given one consonant and one vowel, in how many ways can they be arranged to form
a two-letter word?
c) How many two letter “words” can be written, each containing one consonant and one
vowel, from the alphabet?
12. How many three letter “words” can be written using the letters a, b, c, d, e if repetitions are
allowed but no letter may follow itself?
13. In how many arrangements of letters a, e, i, o, u, x do the five vowels come in alphabetical
order?
14. Given ten consecutive integers, in how many ways can two of them be chosen so that their
sum will be odd?
15
15. How many 5 card hands possible?
16. How many 3 card hands possible of all face cards?
Challenge: In how many ways can 4 people take places for a drive in a 5 passenger car if
a) Any of them can drive?
b) Only 2 of them can drive
16
M328
Chapter 7- Probability
Section 7.6: Warmup
Name: _________________
Date: __________________
1. How many ways can 10 runners place 1st, 2nd, and 3rd?
2. How many ways can 7 people stand in a straight line?
3. Use a counting tree to LIST the ways a penny, quarter, and dime can be flipped, in that
that order.
4. How many ways can the letters of CALCULUS be arranged to form a word?
5. How many ways can you for a 4 digit odd number without repetition from
0,1, 2,3, 4,5,6,7,8,9 ?
17
M328
Chapter 7- Probability
Section 7.6: Permutations and Combinations
Name: _________________
Date: __________________
Permutation: How many ways can you arrange something and Order is important!!
n Pr = P(n, r) denotes the number of permutations of n elements taken r at a time, with r ≤ n.
𝑃(𝑛, 𝑟) =
𝑛!
(𝑛 − 𝑟)!
What happens when you arrange all n objects?
n
Pn = P(n, r) =
Example: How many ways can 10 runners place 1st, 2nd, and 3rd place?
Example: How many ways can 10 runners cross the finish line?
Combination: How many ways can you arrange something and Order is NOT important!!!
𝑛
n C r = C(n, r) or ( 𝑟 ) represents the number of combinations of n elements taken r at a time, with
r ≤ n.
𝑛!
𝑛
𝑛𝐶𝑟 = 𝐶(𝑛, 𝑟) = ( ) =
𝑟
(𝑛 − 𝑟)! 𝑟!
Example: How many ways can 10 people form a committee of 3 people?
Example: How many ways can 10 people form a committee of 10 people?
18
List the 2 letter “words” can you form using the letters E, F, G, and H?
Is this a permutation or combination?
List the 2 letter combinations can you form using the letters E, F, G, and H?
Is this a permutation or combination?
Find the value of the permutation or the combination
a)
5P2
b)
10C10
Distinguishing Between Permutations and Combinations
Permutations
Combinations
Number of ways of selecting r items out of n distinct items
Repetitions are not allowed
Order is important
Order is not important
Arrangements of r items from a set of n items
Subsets of r items from a set of n items
Clue words: arrangement, schedule, order
Clue words: group, committee, sample,
selection
19
A club elects a president, vice-president, and secretary. How many sets of officers are possible if
there are 13 members, any member can be elected to each position, and no person can hold more
than one office?
In a contest in which 9 contestants are entered, in how many ways can 3 distinct prizes be
awarded if no contestant can win more than one prize?
How many ways can 4 kids out of 9 be put in a line?
From 8 names on a ballot, a committee of 3 will be elected to attend a political convention. How
many different committees are possible?
A physics exam consists of 9 multiple-choice questions and 6 open-ended problems in which all
work must be shown. If an examinee must answer 7 of the multiple choice and 2 of the openended problems, in how many ways can the questions and problems be chosen?
To win at LOTTO in a certain state, one must correctly select 6 numbers from a collection of 50
numbers (1 through 50). The order in which the selection is made does not matter. How many
different selections are possible?
20
Your sock drawer contains 8 white socks and 14 blue socks. 4 socks are selected.
a. All 4 socks are white
b. 2 white and 2 blue
c. At least 1 blue sock
A bag contains 7 pennies, 4 nickels, and 5 dimes. Three coins are drawn at random.
a. P(3 pennies)
b. P(2 pennies, 1 nickel)
c. P(3 nickels)
21
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet 7.6B- Permutations and Combinations
Name: _______________________
Permutations and Combinations
1) Calculate the value of the permutation or combination.
a) 8C 3 (by hand)
b)
7P 0
(by hand)
c)
10C 4
d)
19P 3
(on calc)
(on calc)
2) Find C(n, n – 4)
3) In how many ways can horses in a 10-horse race finish first, second, and third?
4) How many different 5 member basketball teams can be formed from 9 players if any
player can play any position?
5) In how many ways can 7 out of 10 monkeys be arranged in a row for a genetics
experiment?
6) In how many ways can 5 salad topping be chosen from 12 selections?
22
7) In an experiment on social interaction, 8 people will sit in 8 seats in a row. In how many
ways can this be done?
8) From a group of 5 men and 12 women, how many committees of 3 men and 5 women
can be formed?
9) How many arrangements of answers are possible in a multiple choice test with 10
questions each of which has 5 possible answers?
10) A baseball team has 20 players. How many 9-player batting orders are possible?
11) How many different samples of 3 apples can be drawn from a crate of 25 apples?
12) In a game of musical chairs, 12 children will sit in 11 chairs that are arranged in a line.
How many seating arrangements are possible?
13) Five cards are marked with the numbers 1, 2, 3, 4 and 5, shuffled, and 2 cards are then
drawn. How many different 2-card hands are possible?
23
14) From 10 names on a ballot, 4 will be elected to a political party committee. In how many
ways can the committee of 4 be formed if each person will have a different
responsibility?
15) A church has 10 bells in its bell tower. Before each church service 3 bells are rung in
sequence. No bell is rung more than once. How many sequences are there?
16) How many ways can five people A, B, C, D, and E, sit in a row at a movie theater if D
and E will not sit next to each other?
17) How many different 3- topping pizzas can be made using sausage, pepperoni, extra
cheese, green peppers, mushrooms, and black olives?
18) You have six different baseball team flags, how many different ways can 3 flags be
placed on a flagpole, one above the other?
19) An urn contains 15 red marbles and 10 white marbles. Five marbles are selected. In how
many ways can the 5 marbles be drawn from the total of 25 marbles:
a. If all marbles are red?
b. If 3 marbles are red and 2 are white?
c. If at least 4 are red marbles?
24
M328- Advanced Algebra 2
Chapter 7- Probability
Name: _______________________
Warmup- Review Permutations and Combinations
In how many ways can a committee of 3 professors be formed from a department of 8
professors?
How many ways can 6 people line up for pictures?
How many ways can a coach select a starting team of one center, two forwards, and 2 guards if
the basketball team consists of 3 centers, 5 forwards, and 3 guards?
Write the formula for nPr.
Write the formula for nCr.
In how many ways can 5 people sit in a row if 2 people have to sit next to each other?
25
M328- Advanced Algebra 2
Chapter 7- Probability
Name: _______________________
Warmup Quiz Review 7.6
1. Find the 5th term of (3x2 – 4y3)9
2. Find C(n, n – 4)
3. A box contains 15 red balls and 10 white balls. Five balls are selected.
a. All balls are red
b. 3 red, 2 white
c. At least 3 red
4. How many numbers greater than or equal to 10 exist, without repetition using the
numbers 0,1, 2,3, 4,5
26
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet 7.6C
Name: _______________________
1. Sally has 7 candles, each a different color. How many ways can she arrange the candles in a
candelabrum that holds 3 candles?
2. Eight toppings for pizza are available. In how many ways can Jim choose 3 of the toppings?
3. How many committees of 5 students can be selected from a class of 25?
4. A box contains 12 black and 8 green marbles. In how many ways can 3 black and 2 green
marbles be chosen?
5. A person playing a word game has the following letters in her tray: QUOUNNTAGGRA.
How many 12-letter arrangements could she make to check if a single word could be formed
from all the letters?
6. Three different hardcover books and five different paperbacks are placed on a shelf. How
many ways can they be arranged if all the hardcover books are together?
27
7. In how many ways can a student choose 4 books from 2 geometry, 4 geography, 5 history and
2 physics books?
8. From a list of 12 books, how many groups of 5 books can be selected?
9. Tom can afford to buy 2 of the 6 CD’s he wants. How many possible combinations could he
buy?
10. A box contains 12 black and 8 green marbles. In how many ways can 5 marbles be chosen?
11. A golf club manufacturer makes irons with 7 different shaft lengths, 3 different grips, and 2
different club head materials. How many different combinations are offered?
12. In how many ways can the 4 call letters of a radio station be arranged if the first letter must
be W or K and no letters repeat?
13. How many 7-digit phone numbers can be formed if the first digit cannot be 0 or 1?
B) What if, in addition, no digit can be repeated?
28
14. A briefcase lock has 3 rotating cylinders, each containing 10 digits. How many numerical
codes are possible?
15. There are 5 different routes that a commuter can take from her home to the office. In how
many ways can she make a round trip if she uses different routes for coming and going?
16. A photographer is taking a picture of a bride and groom together with 6 attendants. How
many ways can he arrange the 8 people in a line if the bride and groom stand together in the
middle?
29
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet- Quiz Review 7.4 and 7.6
Name: _______________________
1. Expand: (3x2 – y)4
2. Find the nth term:
a) Find the 5th term: (3r – 1)6
b) Find the 3rd term of (2k – 1)4
3. Find and show plugged into appropriate formula:
a) P(11, 4)
7
c)  
 2
b) C(18, 5)
4. A baseball manager has 12 players of the same ability. How many 9-player starting lineups
can he create?
5. In how many ways can the letters of the given word be arranged?
a) TRIUMPH
b) ALGEBRA
30
6. A company offers its customers a choice of any 5 toppings for baked potatoes from a list of
20 that are available. How many combinations may be chosen?
7. A pizza company offers 3 sizes of pizzas, 6 toppings and 2 sauce choices. How many
different one-topping pizzas are possible?
8. A board of trustees consists of a chairman, a vice-chairman, a secretary, a treasurer, and 4
members at large. A committee of 3 members is to be selected.
a) How many different committees are possible?
b) How many are possible if the treasurer must be a member?
9. How many even numbers 3 digit numbers can be formed using the numbers: 3, 4, 5, 6, 0?
10. How many ways can 3 out of 7 people be arranged in a row to stand in line for the cafeteria?
11. How many ways can 3 out of 7 people be selected to a homecoming committee, each with
the same responsibilities?
12. How many different 5 digit zip codes are there using the numbers (0-9), if a zip code cannot
start with a 0?
31
M328- Advanced Algebra 2
Name: _______________________
Chapter 7- Probability
Worksheet- 7.7 Odds, Complements, and Sample Space
P(event) =
# of times event occurs
success
=
total
total # of outcomes (same space)
____  P(E)  ____
P(certain event) =
P(impossible event ) =
Complement of an event
P(not event) = _____________
AND
OR
Example: Sock drawer: 19 white, 8 black, 5 blue
P(black)
P(white or blue)
P(purple)
P(not blue)
P(white and black)
Example: Deck or cards
P(black)
P(black 5)
P(red face)
32
Example: Jelly beans: 3 red, 2 blue, 5 black, 1 white, 4 green
P(2 red)
P(2 black)
P(1 red and 1 blue)
P(2 black and 2 green)
Example: Committee of 3: 5 females, 7 males
P(all females)
P(all male)
P(at least 2 males)
Example: Roll 1 die (1-6), what is the probability of getting a prime number 3 times in a row?
Example: Rolling 2 die
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
P(2 fives)
P(sum of 6)
P(sum of 3 or same number)
** Note: there is no overlap… we will discuss overlap later
33
Example: A family of 3 children
Hint: draw a counting tree
P(2 boys)
P(only 1 boy)
P(3 girls)
Probability =
Example: Odds are
Example: P =
success
total
Odds =
5
, what is the probability?
6
5
7
Complement
Odds
Example: Marbles: 6 blue, 4 red
P(red)
Odds(red)
Example: 3 dogs picked: 4 with spots, 3 no spots
Odds (2 spots)
Odds (no spots)
34
success
failures
M328- Advanced Algebra 2
Chapter 7- Probability
Notes- 7.7 Independent vs. Dependent
Name: _______________________
Yesterday’s lesson: 1 GRAB…. 1 or more objects taken at a time  use combinations
 1 grab
 Randomly selected
 r at a time
Today’s lesson: 2 GRABS….successive picks taking 1 at a time
 with replacement
 without replacement
 1 at a time
Flow Chart
35
Dependent (without replacement)
Example: 3 rock CDs, 4 country CDs, 2 Jazz CDs played in succession without replacement.
P(both rock)
P(1 rock, 1 country)
Example: 2 cards selected without replacement
P(both 4)
Example: 3 card hands picked without replacement (in any order)
P(one King, one 4, one 2)
P(exactly 2 queens)
Independent Events (with replacement)
Example: Probability of selecting a face card, replacing it and then selecting an ace.
Example: Find the probability of a baseball team winning the next 3 games if the odds of
winning each game is 5 to 3.
36
M328- Advanced Algebra 2
Chapter 7- Probability
Notes 7.7- Binomial
Name: _______________________
Binomial Probability – is an experiment that consists of repeated independent trials with
only two outcomes in each trial, success or failure. Let the probability of success be p. Then the
probability of failure is 1 – p.
𝑛
( ) 𝑝𝑟 (1 − 𝑝)𝑛−𝑟
𝑟
Three dice are tossed. What is the probability that each occurs?
1. Exactly two 5’s
2. At least two 5’s
3. Three 5’s
4. No 5’s
37
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet 7.7A- Independent and Dependent
Name: _________________
Example: There are 3 nickels, 2 dimes, and 5 quarters in a purse. Three coins are selected in
succession at random.
Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in that order without
replacement.
Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in that order with
replacement.
Find the probability of selecting 1nickel, 1 dime, and 1 quarter in any order with
replacement.
Find the probability of selecting 1 nickel, 1 dime, and 1 quarter in any order without
replacement.
Example: A red, a green, and a yellow die are tossed.
P(all 3 dice show a 4)
P(none of the 3 dice shows a 4)
P(the red die shows an even number and the other 2 show different odd numbers)
P(all 3 dice show the same number)
38
Example: 2 cards picked from a standard deck of cards
P(2 black cards selected without replacement)
P(2 black cards selected with replacement)
P(1 red card and 1 spade in any order without replacement)
P(1 red card and 1 spade in that order selected without replacement)
Example: 3 cards picked from a standard deck of cards
P(exactly 2 queens without replacement in any order)
Example: There are 3 yellow marbles, 2 green and 5 red. Four marbles are selected in
succession at random.
P(1 yellow, 1 green, and 2 red in that order without replacement)
P(1 yellow, 1 green, and 2 red in that order with replacement)
P(1 yellow, 1 green, and 2 red in any order without replacement)
P(1 yellow, 1 green, and 2 red in any order with replacement)
39
Example: A bag contains 7 pennies, 4 nickels, and 5 dimes. Three coins are drawn at random.
P(All pennies or all nickels)
P(exactly 2 dimes)
P(at least 2 dimes)
P(exactly 2 nickels or exactly 2 pennies)
Example: 5 dice are tossed. What is the probability that each occurs?
Exactly two 6’s
No 6’s
All 6’s (Yahtzee!)
40
M328- Advanced Algebra 2
Chapter 7- Probability
Notes 7.7B- Mutually Exclusive or Inclusive Events
Name: _________________
Exclusive Events- 2 events that cannot happen at the same time
Example: When tossing 1 die, what is the probability of getting a 2 or a 3?
Inclusive Events: P(A  B) = P(A) + P(B) – P(A ∩ B)
Example: P(black card or face card)
Example: 2 number cubes
P(sum of 8 or same number)
41
Example: An urn contains 7 white marbles and 5 blue marbles. Four marbles are selected
without replacement.
P(all white or all blue)
P(exactly 3 white)
P(at least 3 white)
P(exactly 3 white or exactly 3 blue)
Example: Two cards are drawn from a standard deck of 52 cards.
P(2 spades)
P(2 spades or 2 red cards)
P(2 red cards or 2 jacks)
P(2 spades or 2 face cards)
42
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet 7.7B – Mutually Exclusive or Inclusive
Name: _________________
Probability of Mutually Exclusive or Inclusive Events
1) The French Club is sponsoring a booth at the school carnival. A player selects a colored
duck from a pond in order to win a prize. The pond contains the following number and
color of ducks: 20 white, 15 gray, 6 red, 3 green, and 1 yellow. You are blindfolded and
asked to select one duck. What is the probability that the duck will be gray or yellow?
2) To win a prize on a game show, a contestant must answer at least 3 true-false questions
correctly out of 5 questions given. What is the probability that the contestant who
guesses on every question will win a prize?
3) A manufacturing company has 2200 employees. The employees attend seminars to
obtain additional training. Suppose that 625 employees will attend the January seminar,
850 will attend the February seminar, and 500 will attend both seminars. What is the
probability that an employee randomly selected will be attending the January or February
seminar?
4) A card is drawn from a deck of 52 cards. Find the probability of drawing a red card or a
face card?
43
5) Of 1560 students surveyed, 840 were seniors and 630 juniors and seniors read a daily
paper. Only 215 of the paper readers were juniors. What is the probability that a student
was a senior or read a daily paper?
6) In one day, 5 different customers bought earrings from the same jewelry store. The store
offers 62 different styles. Find the probability that at least 2 customers bought the same
style?
7) A pair of dice is rolled. What is the probability that the sum of the numbers rolled is
either an even number or a multiple of 3?
8) What is the probability of rolling a die twice and having a 5 and then a 3 show or having
a 2 and then a 4 show?
9) Sophomore club is made up of 12 girls and 9 boys. The club needs to create a committee
with four members. What is the probability that each occurs?
a) All are female or all male
b) At least 3 females
c) Exactly 3 males
d) 2 females and 2 males
44
M328- Advanced Algebra 2
Chapter 7- Probability
Warmup 7.7 Quiz Review
1. Odds =
Name: _______________________
7
9
Probability =
P(not happening) =
2. The odds that Alex will attend a concert each month is
2
. What is the probability that
9
Alex will attend concerts 2 months in a row?
3. Find the probability that 2 cards drawn from a standard deck will both be face or both be
black.
4. There are 13 females and 15 males in your math classroom. What is the probability that
at least 3 males will got to the board if 5 students are selected to go to the board?
5. An experiment consists of rolling a die 10 times. Find the probability that exactly 3 of
the rolls result in a 5. Round 3 decimal places.
6. There are 4 blue, 6 black, 8 white socks. 2 are selected.
a. Both are blue with replacement.
b. Both are blue without replacement.
45
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet Quiz Review 7.7
Name: _______________________
1) If a card is randomly selected from a standard deck of cards.
a) What is the probability of selecting a 5 or 7?
b) What are the odds of this event happening?
c) What is the probability that this event does not happen?
2) Evan has 20 DVD’s. 8 are Curious George, 1 is Dora, 5 are Bob the Builder, and 6 are
Blues Clues. Find the probability that:
a) If 2 DVD’s are picked at random, they are both Bob the builder.
b) If 1 DVD is chosen at random, it is Curious George.
3) 15 friends are picking math classes for next year, 8 choose to take AP Stats. Two friends
are chosen at random. Find the probability that:
a) Both take AP Stats.
b) At least 1 will take AP Stats
46
4) Jen decided that she wanted to bake cupcakes. She decided to vary the frosting. She
made 10 chocolate, 7 vanilla, and 5 with confetti. Find the probability that:
a) Both cupcakes chosen are chocolate with replacement.
b) Both cupcakes chosen are confetti without replacement.
c) If two cupcakes are chosen, find the odds that both have chocolate frosting.
5) There are 7 men and 8 women in a group. If a committee of 6 people is to be made, find
the probability that the committee will have at least 4 men.
6) An experiment consists of tossing a coin 10 times. Find the probability that in exactly 6
of the tosses, the result is tails.
47
M328- Advanced Algebra 2
Chapter 7- Probability
Warmup Review
Name: _______________________
1. Panera offers 3 soups, 5 sandwiches, 4 desserts
a. How many meals possible?
b. How many meals are possible if you pick 2 soups, 3 sandwiches, and 2 desserts
for everyone to share?
2. How many groups of 5 friends can be together if there are 10 friends?
3. How many ways can those 10 friends sit in a 5 passenger car?
4. 3 cards are selected without replacement. Find the probability that exactly two are hearts.
5. A 3 digit number with no repeats using 3, 4, 5, 6, 7, and 8.
a. P(even)
b. P(between 400 and 800)
6. The probability of playing in a game is
4
. What is the probability that the player will
5
play in 7 of the 10 games?
48
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet Conditional Probability
Name: _______________________
Conditional Probability: Is the probability of event A happening if B is known to occur.
P( A | B) 
P( A B)
P( B)
A, B, and C represent three events. Use the following information to calculate the probabilities.
P ( A) 
1
3
P (C | B ) 
P( B) 
3
5
1
4
P(C ) 
P( A  B) 
5
24
2
5
P( A  C ) 
1.) P ( A | C ) 
2.) P( B | A) 
3.) P( Aor B ) 
4.) P (C  B ) 
1
30
The co-ed class has some blondes, brunettes, and some red-haired students, as shown in the
table below. One person is picked at random.
Sophomore
Junior
Senior
Blonde
1
5
6
Brunette
4
6
7
Red-Haired
1
2
4
5.) P(Junior | Blonde)
6.) P(Red-Haired | Senior)
7.) P( Not Brunette | Sophomore)
8.) P(Not Senior | Not Blonde)
49
9.) The probability of wearing a coat, given that it is raining, is 0.62. If the probability of rain
is 0.46, find the probability that you are wearing a coat AND it is raining.
10.) A five digit number is formed from the digits {1,2,3,4,5}. What is the probability that the
number ends in the digits 52, given that it is even? Assume no repetition of digits.
A card is drawn from a standard deck. Find the probabilities
11.) P(Ace | Black) =
12.) P(6 of Clubs | Value of 6)
13.) P( Jack or King | Face Card)
14.) P(Spade | Red)
15.) In Illinois, the probability that a married man drives is 0.90. If the probability that a
married man and his wife both drive is 0.85, what is the probability that his wife drives given
that he drives?
16.) Brian often speeds while driving to school in order to arrive on time. The probability that
he will speed to school is 0.75. If the probability that he speeds and gets stopped by a
Schaumburg police officer is 0.25, find the probability that he is stopped, given that he is
speeding.
50
17. The managers of a corporation were surveyed to determine the background that leads to a
successful manager. Each manager was rated as being either good, fair, or poor manager by
his/her boss. The manager’s educational background was also noted. The data appears below.
Rating
Good
Fair
Poor
Totals
HS Degree
7
8
2
17
Some College
3
15
6
24
College Degree
25
45
1
71
Master/PHD
4
19
25
48
Total
39
87
34
160
a. Given that a manager is only a fair manager, what is the probability that this manager has
no college background?
b. Given that a manger is a good manager, what is the probability that this manager has
some college background?
c. Given that a manger has a PhD, what is the probability that the manager was ranked
poor?
18. A local country club has a membership of 600 and operates facilities that include an 18-hole
championship golf course and 12 tennis courts. Before deciding whether to accept new
members, the club president would like to know how many members regularly use each facility.
A survey of the membership indicates that 62% regularly use the golf course, 43% regularly use
the tennis courts, and 4% use neither of these regularly. Given that a randomly selected member
uses the tennis courts regularly, find the probability that they also use the golf course regularly.
51
19. Six of your eight male classmates practice tai-chi, and nine of your thirteen female
classmates practice tai-chi. If one classmate is picked at random, find P(male│practice
tai-chi).
5
20. The probability of waking up late, given that your alarm clock breaks is 8. If the
3
probability that your alarm clock breaking is 16, find the probability that you wake up late
and you alarm clock breaks.
21. You have 51% chance of getting accepted to clown college. You have a 33% chance of
getting accepted to clown college and getting the esteemed Bozo Scholarship. What is
the probability of getting the Bozo Scholarship, given you have already been accepted to
clown college?
3
22. The probability of getting a circus job, given that you flunk out of clown college is 10. If
1
the probability of getting a circus job and flunking out of clown college is 24, what is the
probability of flunking out of clown college?
23. A survey taken at Schaumburg High School showed that 48% of respondents liked
soccer, 66% liked basketball, and 38% liked hockey. Also, 30% like both soccer and
basketball, 22% liked basketball and hockey, and 28% liked soccer and hockey. Finally
12% liked all 3 sports.
a. P(a student likes soccer│he/she likes basketball)
b. P(a student likes hockey and basketball │he/she likes soccer).
52
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet Venn Diagram
Name: _______________________
A Venn Diagram is made up of two or more circles. It is often used in probability to show
relationships between events.
Rectangle ‘S’ represents a sample space of possible outcomes.
Circles ‘A’ and ‘B’ each represent specific unique events in the sample space S.
If it is possible for two events A and B to both happen at the same time, then their circles will
intersect on a Venn Diagram.
I. Shade the region of the picture corresponding to each listed event.
1) A
2) B
S
S
A
A
B
3) 𝑨 ∩ 𝑩
B
4) 𝑨 ∪ 𝑩
S
S
A
A
B
53
B
The complement of an event ‘A’ refers to all outcomes that are NOT included in ‘A’.
The notation for complement is 𝑨𝒄 . The probability of 𝑨𝒄 = 1 – P(A).
Shade the region corresponding to each listed event.
5) 𝑨𝒄
6) 𝑩𝒄
S
S
A
A
B
7) (𝑨 ∩ 𝑩)𝒄
B
8) (𝑨 ∪ 𝑩)𝒄
S
S
A
A
B
B
Let ‘S’ represent rolling a 6-sided die. A and B represent different events. Draw Venn diagrams
for each pair of events below.
9)
A = number less than 3
B = an even number
10)
A = a ‘2’
B = an even number
11)
A = a ‘2’
B = a ‘3’
12)
A = a # greater than or equal to ‘3’
B = a # less than or equal to ‘5’
13) In which problem (#9 – 12) are the events mutually exclusive?
54
II. 80 students were asked if they take Math and History classes. The Venn diagram shows the
number of students who took each class. Suppose one student is chosen at random, and answer
the questions below.
Students in Math
Students in History
23
14
26
17
14) P(student takes Math) =
15) P(student takes History) =
16) P(student takes Math AND History) =
17) P(student takes Math OR History) =
18) What does the ‘17’ represent in the diagram above?
19) Are the events “taking Math class” and “taking History class” independent? Hint: use the
formula 𝑷(𝑨 ∩ 𝑩) = 𝑷(𝑨) ∙ 𝑷(𝑩) for independent events to decide.
55
III.
A
B
C
D
Passing Photography
Passing Economics
20) Which regions (A,B,C, or D) represent passing both classes?
21) Which regions represent failing both classes?
22) Which regions represent passing exactly one class?
Students’ records indicate that the probability of passing photography is 0.75, the
probability of failing economics is 0.65, and the probability of passing at least one of the
two courses is 0.85. Regions A, B, C, and D make up the entire sample space of students
taking both classes.
23) P(passing economics) =
24) P(passing both courses) =
25) P(failing both courses) =
26) P(passing exactly one course) =
56
IV. 100 women who recently gave birth were given a survey with 3 yes/no questions, and the
following information was recorded.
The mother…
Is over 35 years of age
Just had her first child
Has a career
Is over 35 and just had first child
Has a career and just had first child
Is over 35 and has a career
Is over 35, has a career, and just had first child
Number of Respondents
27
21
42
17
9
16
7
Use the Venn Diagram to organize the information.
If a mother from this group is randomly selected, find the probabilities below.
27) P(replied ‘yes’ to exactly 2 of the 3 questions above)
28) P(replied ‘yes’ to at least 2 of the 3 questions above)
29) P(replied ‘yes’ to at least one of the 3 questions above)
30) P(replied ‘yes’ to none of the 3 questions above)
57
17% of students take calculus, statistics and go to prom.
3% take calculus, statistics, and do not go to prom.
24% take calculus and go to prom.
21% take statistics and go to prom.
53% go to prom.
13% of statistics students do not go to prom or take calculus.
10% of students do not take statistics, do not take calculus, and don’t go to prom.
Fill in the venn diagram below.
31. Given the student takes calculus what is the probability they will attend prom?
32. What is the probability that the student takes calculus or goes to prom?
33. Given the student does not take statistics, what is the probability the student is going to
prom?
34. Given the student is going to prom, what is the probability they are not taking statistics?
58
Fifteen students go to the movies. The theater is playing Star Wars, Star Trek, and Harry Potter.
Each student wants to see at least one of the movies. Three of the students will only see Star
Wars. Two students refuse to see anything except Star Trek. Four students would be willing to
see any movie featuring space travel. Three students don’t care what movie is playing, they will
see anything. In all, a total of 8 students would see Star Wars, and a total of 7 students would see
Star Trek.
35) What is the probability that a person will only see Harry Potter?
36) What is the probability that a person is willing to see Star Trek or Harry Potter?
37) Given the student will not see Star Wars, what is the probability they will see Star Trek?
59
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet Test Review #1
Name: _________________
1. How many five-digit numbers (without repetition) are possible if:
a) Only odd digits can be used?
b) The number must be even?
c) Repetition is allowed with no other restrictions?
2. How many even numbers of 3 digits can be written using 0, 1, 2, 4, 5, 7 without
repetition?
3.
In how many ways can 4 people take seats in a row of 9 chairs?
4.
In how many ways can a constant and then a vowel be chosen from the letters p,a, t, e, s,
q?
5. Out of 8 managers and 13 assistants a committee consisting of 3 managers and 6
assistants is chosen. In how many ways can this be done if:
a) Any manager and any assistant can be included?
b) One particular manager must be on the committee?
60
c) Three particular assistants must be on the committee and one particular manager
cannot be on the committee?
6. How many different 5 card hands containing exactly 3 aces can be dealt?
7. Simplify:
a)
8.
𝑎!
(𝑎−2)!
b)
(𝑥+2)!
𝑥!
In science class, 3 of 8 girls wear glasses and 4 of 12 boys wear glasses. What is the
probability of choosing a boy or a person who wears glasses?
9. A bag of candy contains 4 grapes, 3 lemons and 6 oranges. What is the probability of
choosing:
a) A grape or an orange?
b) A grape, followed by a lemon (without replacement)?
c) Two oranges (one after the other, without replacement)?
10. Two different letters are chosen at random from the word PIANO. What is the
probability that both are vowels?
61
11. A three digit number with no repeated digits is made from the digits 2 through 8. What is
the probability that the number is:
a) Even?
b) Between 400 and 700?
12. A coin is tossed 5 times. What is the probability of getting:
a) All heads?
b) Exactly 2 heads?
c) At least 1 tail?
13. There are 8 used and 12 new cars in a dealer’s lot. What is the probability of selling 2
used and 3 new cars, if 5 of them are sold at random?
14. Write the first 3 terms of:
(𝑥 4 − 2𝑦)11
15. Find the sixth term of: (3𝑥 + 𝑦)8
62
M328- Advanced Algebra 2
Chapter 7- Probability
Warmup- Review
Name: _________________
1. Find the 8th term of (3x + 4y)10
2. Expand (3x – y)4
3. Solve for n… show all work!
(n  2)!
6
n!
4. How many ways can 6 different books be placed on a shelf if the dictionary must be
placed on the end?
5. How many different 3 digit odd numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6?
6. How many different orders can 4 people be selected to get their homework checked out
of 8 students?
7. 6 out of 8 freshman students are in an elective class. 7 out of 10 sophomores are in an
elective class. 8 out of 13 juniors are in an elective class. 9 out of 10 seniors are in an
elective class. If you pick a student at random P(student is a junior | not in an elective)
63
M328- Advanced Algebra 2
Chapter 7- Probability
Worksheet Test Review #2
Name: _________________
1. Mary’s car will not start. Based on past experience, Mary knows that the probability that it is
the battery is 0.65, the probability that it is the automatic choke on the carburetor is 0.45, and the
probability that it is both is 0.24. Find the probability that it is either of these two problems.
2. Georgia is repairing her car. She has removed the 6 spark plugs. Four are good and two are
defective. She now selects one plug and then, without replacing it, selects a second plug. What
is the probability that both spark plugs selected are good?
3. The Casco Corp. uses many different delivery services. The probability that any given parcel
will be sent with the ABC Speedy Delivery Service is 0.71. The probability that the parcel will
arrive on time given that the ABC Speedy Delivery was used is 0.93. If a parcel is randomly
selected, find the probability that it will be sent with the ABC Speedy Delivery Service and that
it will arrive on time.
4. Jon has ten single dollar bills of which three are counterfeit. If he selects four of them at
random, what is the probability of getting two good bills and two counterfeit bills?
5. Two fair dice are rolled. Find the probability that:
a) the sum is 5.
64
b) the sum is 6 or both dice show the same number.
c) the sum is even
6. Three fair coins are tossed. Find the probability that:
a) all 3 are heads.
b) exactly two are heads.
7. A nursing home employs 13 nurses on its morning shift and 11 nurses on its afternoon shift.
As a cost-reducing measure, management decides to fire 2 nurses. What is the probability that:
a. both fired nurses will be from the morning shift?
b. both fired nurses will be from the afternoon shift?
c. one of the fired nurses will be from the morning shift and one will be from the
afternoon shift?
8. Solve for n:
(n + 1)! = 6 ∙ n!
65
9. Expand each of the following:
a) (x + y)5
b) (x + y5)4
c) (2x – y)4
9. Find the indicated term:
a) 4th term of (a + b)11
b) 3rd term of (x + 3y)8
c) 6th term of (x – y)12
d) 5th term of (x2 – y4)10
66