Name: _______________________________________ Date: ______________
1
Algebra 2/Trig: Chapter 16 Probability In this unit, we will…
Determine whether a situation is modeled by a permutation or a combination, or a factorial
Generate and use Pascal’s Triangle to determine combinations
Determine exact probabilities that are modeled by Bernoulli trials
Determine “at least” and “at most” probabilities that are modeled by Bernoulli trials
Name: _______________________________________ Date: ______________
2
Day 1: Algebra2/Trig: Chapter 16-2: Permutations and Combinations Let’s assume there are 25 students in this class. If I were to pick a President, a Vice-President, and a Secretary of the class AT RANDOM,
= ______ total ways. # ways to choose a
P
#ways to choose
VP
#ways to choose S
A permutation is an arrangement of people or things where the ORDER MATTERS. The notation nPr means you are taking n objects and you are arranging r of them. When you’re writing a permutation, you start with the “n,” and count down towards 1, multiplying the whole time, r number of times.
nPr = ( ) ( ) ( )
r times
Meaning The notation The math Arranging 4 objects from 10 10P4
Arranging 2 objects from 52
Arranging 4 objects from 4
The last case is a special case of a permutation, where you are arranging ALL of the objects. In nPr, if n=r, this is called a factorial. Factorial is represented by an exclamation point.
10! = 10P10 = = a really big number.
Meaning The notation The math Arranging 5 objects from 5 5P5 or 5!
Arranging 6 objects from 6
Name: _______________________________________ Date: ______________
3
A combination is group of people or things where the ORDER DOESN’T MATTER.
You don’t have to memorize that formula… your calculator will do all of it! Your calculator can do all of this from the MATH PRB menu. Example: 10P4 type 10 MATH PRB 2 ENTER 4 ENTER Example: 5! type 5 MATH PRB 4 ENTER 4 ENTER Example: 10C4 type 10 MATH PRB 3 ENTER 4 ENTER
Type Words to Look For Sample Problem Answer
Permutation (Lining up different objects)
Line up Arrangement Put in Order Rank
# of ways to line up 3 different objects out of 10
10P3 = = 720
Factorial Same as a permutation, but you’re lining up or ordering ALL of the objects.
Same as permutation
# of ways to line up all 5 objects out of 5
5P5 = 5! =
Combination (taking a group of objects)
Group Committee Team
# of ways to group 3 objects out of 10
10C3 =
=120
Problem Permutation or
Combination? Write in nPr, nCr, or n! notation and use calculator to answer
1. The # of ways to line up 10 different chairs
2. The # of ways to line up 4 out of 10 different chairs
3. The # of ways to pick a group of 3 chairs from 10
4. # of ways to pick a president, a VP, and a treasurer from the students in this class.
5. # of ways to pick a committee of 3 people from this class
6. # of ways to line up all the people in this class.
7. # of ways to choose a 3-scoop sundae from 31 flavors of Baskin Robbins
8. # of ways to make a 3-scoop cone from 31 flavors if you care what order the scoops are in
Name: _______________________________________ Date: ______________
4
You try it! Special Case: Probability with repetition: Example: How many ways are there to arrange the letters of the word “Mississippi?” Every time there is a letter that repeats, count how many times each letter repeats, divide the answer by that number factorial. Solution: There are 11 letters in “Mississippi”. The “I”s repeat 4 times, the “s” repeats 4 times and the “p” repeats twice.
1. Which formula can be used to determine the total number of different eight-letter arrangements that can be formed using the letters in the word DEADLINE? 1) 2)
3)
4)
2. Which expression represents the number of different 8-letter arrangements that can be made from the letters of the word "SAVANNAH" if each letter is used only once? 1)
2)
3)
4)
Name: _______________________________________ Date: ______________
5
3. 4. How many different three-member teams can be selected from a group of seven students? 1) 1 2) 35 3) 210 4) 5,040
5. If there are four teams in a league, how many games will have to be played so that each team plays every other team once? 1) 6 2) 8 3) 3 4) 16
6. How many different five-member teams can be made from a group of eight students, if each student has an equal chance of being chosen? 1) 40 2) 56 3) 336 4) 6,720
7. 7. Alan, Becky, Jesus, and Mariah are four students in the chess club. If two of these students will be selected to represent the school at a national convention, how many combinations of two students are possible?
8. An algebra class of 21 students must send 5 students to meet with the principal. How many different groups of 5 students could be formed from this class?
9. 9. Megan decides to go out to eat. The menu at the restaurant has four appetizers, three soups, seven entrees, and five desserts. If Megan decides to order an appetizer or a soup, and one entree, and two different desserts, how many different choices can she make?
10. On a bookshelf, there are five different mystery books and six different biographies. How many different sets of four books can Emilio choose if two of the books must be mystery books and two of the books must be biographies?
CHALLENGE:
Name: _______________________________________ Date: ______________
6
SUMMARY Exit Ticket
Name: _______________________________________ Date: ______________
7
Homework: pp. 685-686 #3-21eoo, 23,27,29,31, 35, 47 Page 685
Name: _______________________________________ Date: ______________
8
Page 686
Name: _______________________________________ Date: ______________
9
Answer Key
Name: _______________________________________ Date: ______________
10
Day 2: Algebra2/Trig: Permutations and Combinations Review Warm - Up:
Example 1: The letters of the word CABIN are rearranged at random. What is the theoretical
probability that one arrangement chosen at random will begin and end with a vowel?
Example 2: The letters of the word SEED are arranged at random. What is the probability that
the arrangement begins and ends with E?
Combinations and Probability
Example 1: A variety box of instant oatmeal contains 10 plain, 6 maple, and 4 apple cinnamon
flavored packets. Sara reaches in and takes 3 packets without looking. Find each probability:
a) P(2 plain, 1 non-plain)
b) P(1of each flavor)
c) P(2 plain, 1 maple)
Name: _______________________________________ Date: ______________
11
1. What is the probability of rolling a 2 on a toss of a die?
2. What is the probability of choosing a king when a card is drawn out of a deck of 52 cards?
3. An urn contains 3 red marbles, 5 green marbles and 4 orange marbles. If two marbles are drawn
what is the probability of drawing :
a. One red and one orange marble with replacement?
b. One red and one orange marble without replacement?
4. Evaluate: 5!
5. Evaluate: 7!
6. a) How many different ways can the letters STRAWBERRIES be arranged?
b) How many different arrangements begin and end with an R?
c) What is the probability that the arrangement will begin and end with an R?
7. There are 3 seniors and 15 juniors in Mr. Cameron’s math class. Three students are chosen at
random from the class. What is the probability that the group consists of 1 senior and 2 juniors?
8. If the probability that it is going to rain is 2/5, what is the probability that it will not rain?
Name: _______________________________________ Date: ______________
12
9. Evaluate: 5P2
10. Evaluate: !3
!6
11. Evaluate: 6C4
12. Mr. Jones a choosing a committee of 4 students from a group of 7 students.
a. How many different committees can be formed?
b. How many different committees can be formed if each student is assigned a job:
president, vice president, treasurer and secretary?
13. Lucy is choosing three members of the national honor society to travel to Maryland. How many
different committees can be formed if she has 8 students to choose from?
14. Evaluate: 8P5
15. Evaluate: 7C 3
16. How many different 3 person committees can be formed from 6 students?
Name: _______________________________________ Date: ______________
13
17. a) How many different ways can the letters MATHEMATICS be arranged?
b) How many different ways can the letters MATHEMATICS be arranged if the arrangement must
begin with a vowel?
c) What is the probability that the arrangement will begin with a vowel?
18. a) How many different ten letter arrangements can be made from the letters ASSESSMENT?
b) How many of these ten letter arrangements can be made if it begins and ends with an S?
c) What is the probability that the arrangement will begin and end with an S?
19. a) How many different ways can the letters MASSACHUSETTS be arranged?
b) How many different ways can the letters MASSACHUSETTS be arranged if it must begin with an
S?
c) What is the probability that the arrangement will begin with an S?
20. A committee of 5 is to be chosen from a group of 8 men and 6 women. What is the probability
that the committee formed consists of 2 men and 3 women?
Name: _______________________________________ Date: ______________
14
Day 3: Algebra2/Trig Chapter 16-2: Bernoulli Binomial Probability: “EXACTLY” Probability is a very complicated subject. There is a simple situation called a “Bernoulli Trial,” named after the famous mathematician Daniel Bernoulli. A Bernoulli trial is when the only two outcomes is something that can be defined as a “success” and the opposite of that occurring is called a “failure,” and those two probabilities add up to 1.
Example 1: A fair coin is tossed 5 times. What is the probability that it lands tails up exactly 3 times?
(1) ( )1
2
3 (2) 101
2
5( ) (3) 3
5 (4) 10
1
2
3( )
Solution: n= r= p= q=
Name: _______________________________________ Date: ______________
15
Example 2: The probability that Kyla will score above a 90 on a mathematics test is
. What is the
probability that she will score above a 90 on three of the four tests this quarter?
(1) 4 3
3 14
5
1
5C ( ) ( ) (2)
3
4
4
5
1
5
3 1( ) ( ) (3) 4 3
1 34
5
1
5C ( ) ( ) (4)
3
4
4
5
1
5
1 3( ) ( )
Solution: n= r= p= q= Problems:
1. If the probability that the Islanders will beat the Rangers in a game is
, which expression
represents the probability that the Islanders will win exactly four out of seven games in a series against the Rangers?
(1) ( ) ( )2
5
3
5
4 3 (2) 7 4
4 32
5
2
5C ( ) ( ) (3) 5 2
2 34
7
3
7C ( ) ( ) (4) 7 4
4 32
5
3
5C ( ) ( )
2. Which fraction represents the probability of obtaining exactly eight heads in ten tosses of a fair coin?
(1) 45
1 024, (2)
90
1 024, (3)
64
1 024, (4)
180
1 024,
3. Mr. and Mrs. Doran have a genetic history such that the probability that a child being born to them
with a certain trait is
. If they have four children, what is the probability that exactly three of their
four children will have that trait?
Name: _______________________________________ Date: ______________
16
4. If the probability that it will rain on any given day this week is 60%, find the probability it will rain
exactly 3 out of 7 days this week.
5. Jim can drive a golf ball over 220 yards 40% of the time. He regularly plays on a golf course where drives of that distance are needed on 12 holes. Determine the probability that exactly 7 of his drives will be over 220 yards.
6. If a fair coin is tossed five times, the probability of getting exactly two heads is
1) 2) 3) 4)
7. The probability that Laura wins a tennis match against Jennifer is . What is the probability that Laura wins exactly three of the next four matches she plays against Jennifer?
1) 2) 3) 4) 8. In basketball, Nicole makes 4 baskets for every 10 shots. If she takes 3 shots, what is the probability that exactly 2 of them will be baskets? 1) 0.288 3) 0.600 2) 0.432 4) 0.960
Name: _______________________________________ Date: ______________
17
FOR BERNOULLI EXACTLY TRIAL: the VERB you are doing. Example: Flipping ONE coin SUCCESS: What outcome is a success to you. Depends on how the problem is phrased. Example: “Getting a Head” on a coin flip FAILURE: The opposite of your success. Quite literally, in this example, “NOT getting a Head” is a failure. PROBABILITY OF SUCCESS: probability of the even happening ONE TIME. Example: P(head) = ½ # of successes: the number of times the problem is asking for a “success” to happen PROBABILITY OF FAILURE: P(success) + P(failure) = 1
SUMMARY: Exit Ticket
Name: _______________________________________ Date: ______________
18
Homework
Page 699: #4, 7, 8, 11 Answer Key
Name: _______________________________________ Date: ______________
19
Page 706: #3 – 5 (a and c only), 18, 19
Name: _______________________________________ Date: ______________
20
Day 4 - Algebra2/Trig Bernoulli Trials: AT LEAST or AT MOST If the Bernoulli problem says “at least” or “at most,” you have to add up all of the “exactly” that represent the problem. At least: At most:
Example One: If the probability of rain is 20% on any given day this week, then what is the probability that it will rain on at least 3 days of the next 5? SOLUTION: Example 2:
On any given day, the probability that the entire Watson family eats dinner together is
. Find the
probability that, during any 7-day period, the Watsons eat dinner together at most two times.
=
Name: _______________________________________ Date: ______________
21
1. Team A and team B are playing in a league. They will play each other five times. If the probability
that team A wins a game is
, what is the probability that team A will win at least four of the five
games? 2. Dave is the manager of a construction supply warehouse and notes that 60% of the items
purchased are heating items, 25% are electrical items, and 15% are plumbing items. Find the probability that at least three out of the next five items purchased are heating items.
3. Tim Parker, a star baseball player, hits one home run for every ten times he is at bat. If Parker
goes to bat five times during tonight’s game, what is the probability that he will hit at most 1 home runs?
4. The probability that a planted watermelon seed will sprout is
If Peyton plants seven seeds from
a slice of watermelon, find, to the nearest ten thousandth, the probability that at least five will sprout.
5. On mornings when school is in session in January, Sara notices that her school bus is late one-
third of the time. What is the probability that during a 5-day school week in January her bus will be late at most three times?
Name: _______________________________________ Date: ______________
22
6. The probability of rain on the last day of July is 90%. If the probability remains constant for the
first seven days of August, what is the probability that it will rain at least six of those seven days in August?
7. East West Airlines has a good reputation for being on time. The probability that one of its flights
will be on time is .91. If Mrs. Williams flies East West for her next five flights, what is the probability that at most three of them will be on time? Round your answer to the nearest thousandth.
8. Dr. Glendon, the school physician in charge of giving sports physicals, has compiled his
information and has determined that the probability a student will be on a team is 0.39. Yesterday, Dr. Glendon examined five students chosen at random. Find, to the nearest hundredth, the probability that at least four of the five students will be on a team.
9. When Joe bowls, he can get a strike (knock down all the pins) 60% of the time. How many times
more likely is it for Joe to bowl at least three strikes out of four tries as it is for him to bowl zero strikes out of four tries? Round your answer to the nearest whole number.
10. As shown in the accompanying diagram, a circular target with a radius of 9 inches has a bull’s-eye
that has a radius of 3 inches. If five arrows randomly hit the target, what is the probability that at least four hit the bull’s-eye?
Name: _______________________________________ Date: ______________
23
11. Dave does not tell the truth
3
4 of the time. Find the probability that he will tell the truth at most twice out of the next five times.
12. A board game has a spinner on a circle that has five equal sectors, numbered 1, 2, 3, 4, and 5, respectively. If a player has four spins, find the probability that the player spins an even number no more than two times on those four spins.
SUMMARY: If the probability of rain is 20% on any given day this week, then what is the probability that it will rain on at least 3 days of the next 5? SOLUTION:
=
Name: _______________________________________ Date: ______________
24
Day 5 - Algebra2/Trig: Pascal’s Triangle and Binomial Expansion
Pascal’s Triangle
The basics of expanding binomials: ( ) because anything to the 0 power is 1. ( ) because anything to the 1st power is itself. ( ) by FOIL But, this only works for two binomials. It doesn’t work for more than two! Another way is like long multiplication:
If we want to expand ( ) we can multiply the result of ( ) by another We can keep going on like this, but I’ll spare you the work:
( ) There is a pattern in the a’s. There is a pattern in the b’s. There is a pattern in the coefficients.
( )
( ) ( ) ( ) ( )
Name: _______________________________________ Date: ______________
25
To expand a binomial of the form ( )
Write out powers of a. The powers of a start at ____ and ___________.
Write out powers of b. The powers of b start at ____ and ___________.
Write out the coefficients. These come from Pascals’s triangle, OR you can use
combinations starting at ____C___.
Determine the following binomial expansions:
(x – 2) 4
(1 – b) 3
(2 + a2
) 2
(2a + 3) 3
Name: _______________________________________ Date: ______________
26
1. What is the third term in the
expansion of ?
1)
2)
3)
4)
2. What is the coefficient of the fifth
term in the expansion of ?
1) 8 2) 28 3) 56 4) 70
3. The fourth term in the expansion of
is
1)
2)
3)
4)
4. What is the third term in the
expansion ?
1)
2)
3)
4)
5. What is the fifth term in the
expansion of ? 1) 2) 3) 4)
6. What is the third term in the
expansion of ?
1) 2) 3) 4)
7. What is the fourth term in the
expansion of ?
1) 2) 3) 4)
8. Find, in simplest form, the middle
term in the expansion of .
9. The third term in the expansion of
is
1) 2) 3) 4)
10. Expand and express in simplest
form:
Name: _______________________________________ Date: ______________
27
Homework: pg. 711 #6, 7, 9, 12, 14, 16
Page 711
Name: _______________________________________ Date: ______________
28
Probability Review
1)
2)
3)
Name: _______________________________________ Date: ______________
29
4)
5)
6)
Name: _______________________________________ Date: ______________
30
7)
8)
9)
Name: _______________________________________ Date: ______________
31
10)
11) There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send
four representatives to the State Conference.
a.) How many different ways are there to select a group of four students to attend the conference?
b.) If the members of the club decide to send two juniors and two seniors, how many different
groupings are possible?
Name: _______________________________________ Date: ______________
32
12) The local Family Restaurant has a daily breakfast special in which the customer may choose
one item from each of the following groups:
Breakfast Sandwich Accompaniments Juice
egg and ham egg and bacon egg and cheese
breakfast potatoes apple slices
fresh fruit cup pastry
orange cranberry
tomato apple grape
a.) How many different breakfast specials are possible?
b.) How many different breakfast specials without meat are possible?
13) A family consists of 3 children. What is the probability that at most 2 of the children are boys?
Name: _______________________________________ Date: ______________
33
A2T Probability Review #2
1.
2.
3.
4.
Name: _______________________________________ Date: ______________
34
5.
6. What is the fifth term in the expansion (a +bi)7?
a. 35a3b
4 b. -21a
2b
5i c. -35a
3b
4 d. 21a
2b
5i
7.
8. The Hertel family is traveling to Aruba this July. They will be staying for 8 days. The
probability that it will rain on any given day in Aruba is 1/5. What is the probability that it will
rain for at most 2 days on their vacation?
9.
Name: _______________________________________ Date: ______________
35
10.
11.
12.
13.
Name: _______________________________________ Date: ______________
36
14.
15.
Mrs. Hertel would like to form a four person math team from a group of 12 students. How
many different teams can be chosen?
16.
Mrs. Butterworth has 16 friends that would like to travel the world with her to sell syrup. How
many different groups of 5 can she take on her journey?
17. a) How many different 6 letter arrangements of BANANA can be formed?
b) How many of these 6 letter arrangements will begin with a vowel?
c) What is the probability that the arrangement will begin with a vowel?
18.
Name: _______________________________________ Date: ______________
37
19. Expand: (2x + y )5
20. A committee of 5 is to be chosen from group of 3 freshmen, 4 sophomores, 6 juniors and 7
seniors.
What is the probablity that the committee chosen will contain 1, freshman, 1 sophomore, 1 junior
and 1 senior?
