Honors Advanced Algebra with Trigonometry
Probability
Target Goals
By the end of this chapter, you should be able to…
- Solve problems using the Counting Principle. (Day 1)
_____ got it_____needs work_____ no clue
- Solve problems involving permutations. (Day 2)
_____ got it_____needs work_____ no clue
- Solve problems involving combinations. (Day 3)
_____ got it_____needs work_____ no clue
- Find the probability of an event and determine the odds of success or failure. (Day 4)
_____ got it_____needs work_____ no clue
- Find the probability of two or more independent or dependent events. (Day 5)
_____ got it_____needs work_____ no clue
- Find the probability of mutually exclusive events or inclusive events. (Day 6)
_____ got it_____needs work_____ no clue
Honors Advanced Algebra with Trigonometry
Assignment Guide
Probability
The Counting Principle
Target Goal:Solve problems using the Counting Principle.
.
HW #1Worksheet
Linear Permutations
Target Goal:Solve problems involving permutations.
HW #2Worksheet
Combinations
Target Goal: Solve problems involving combinations.
HW #3Worksheet
Day 1 – Day 3 Review
HW #3.5Worksheet
Probability
Target Goal: Find the probability of an event and determine the odds of success or failure.
HW #4Worksheet
QUIZ Day 1 – Day 3
Multiplying Probabilities
Target Goal: Find the probability of two or more independent or dependent events.
HW #5Worksheet
Adding Probabilities
Target Goal: Find the probability of mutually exclusive events or inclusive events.
HW #6Worksheet
QUIZ Day 4 – Day 5
Probability Review
HW #7Worksheet
Tentative Test Date: ______
Honors Advanced Algebra with TrigonometryName: ______
The Counting PrincipleDate: ______
Probability Notes (Day 1)Period: ______
Target Goal: Solve problems using the Counting Principle.
Kevin wants to buy a new car. He has already chosen the make and model, but still must decide between automatic or manual transmission, a sunroof or no sunroof, and a color (red, white, or black). These decisions are all called INDEPENDENT EVENTS because one decision will not affect the other. We can use a tree diagram to illustrate all of the choices that Kevin still must make:
Now we can see that Kevin has 12 choices to choose from. We could list these choices in a SAMPLE SPACE:
The total number of choices can also be found without drawing a diagram:
This application is an example of the BASIC COUNTING PRINCIPLE which states the following:
Suppose an event can occur in p different ways; another event can occur in q different ways, then there are ways both events can occur.
Ex 1:How many 4-letter patterns can be formed using the letters u, v, w, x, y, and z if the letters may be
repeated?
But what if the letters in the example above cannot repeat? This would be a DEPENDENT EVENT becausethe number of choices for one event does affect other events.
Ex 2: How many 4-letter patterns can be formed using the letters u, v, w, x, y, and z if letters may not be
repeated?
Practice:
1. Tell whether the events are independent or dependent, then solve.
a. How many 2-digit numbers can be formed from 1, 2, 3, 4, and 5, if repetitions are allowed?
b. How many ways can 5 different books be arranged on a shelf?
c. How many seven-digit numbers can be formed from the digits 0 - 9 if each digit can be used more
than once?
d. From the letters a, e, i, o, r, m, and n, how many different four letter patterns can be formed if no
letter occurs more than once?
2. From 1d above, how many of these patterns begin with a vowel and end with a consonant?
3. List the possible outcomes if you flip a coin three times. What is this list called?
4. How many ways can 5 cars be parked along the street if the only SUV must be in the middle?
5. How many ways can six books be arranged on a shelf if one of the books is a dictionary and it must be
on an end?
6. Using the letters from the word equality, how many four letter patterns can be formed in which q is followed
immediately by u?
7. Draw a chart to represent the sample space of possible outcomes for rolling two dice.
Assignment #1: Worksheet
Honors Advanced Algebra with TrigonometryName: ______
The Counting PrincipleDate: ______
Assignment #1Period: ______
Solve each problem.
1. The letters g, h, j, k, and l are to be used to form 5-letter passwords for an office security system. How many passwords can be formed if the letters can be used more than once in any password?
2. A store has 15 sofas, 12 lamps, and 10 tables at half price. How many different combinations of a sofa, a lamp, and a table can be sold at half price?
Draw a tree diagram to illustrate all the possibilities.
3. the possibilities for boys and girls 4. the possibilities for boys and girls
in a family with two children in a family with three children
Solve each problem.
5. A license plate must have two letters (not I or O) followed by three digits. The last digit cannot be zero. How many possible plates are there?
6.There are five roads from Albany to Briscoe, six from Briscoe to Chadwick, and three from Chadwick to Dover. How many different routes are there from Albany to Dover via Briscoe and Chadwick?
7.For a particular model of car, a car dealer offers 6 versions of that model, 18 body colors, and 7 upholstery colors. How many different possibilities are available for that model?
8.How many ways can six different books be arranged on a shelf?
9.Three different colored six-sided dice are tossed. How may distinct outcomes can occur?
10. How many ways can six books be arranged on a shelf if one of the books is a dictionary and it must be on
an end?
11. Using the letters from the word equation, how many 5-letter patterns can be formed in which q is followed
immediately by u?
12. Consider the letters a, e, i, o, r, s, and t.
a. How many different 4-letter patterns can be formed from these letters if no letter occurs more
than once?
b. How many of these patterns begin with a vowel and end with a consonant?
13. How many 5-digit numbers exist between 65,000 and 69,999 if no digit is to be repeated in each number?
14. A Chinese restaurant offers a special price for customers who dine before 6:30 p.m. This offer includes an
appetizer, a soup, and an entrée all for $6.95. There are 4 choices of appetizers, 3 soups, and 5 entrées.
How many different meals are available under this offer?
15. Four ferry boats run round trips between Harrod and Lafayette.
a. How many different ways can a traveler make a round trip?
b. How many different ways can a traveler make a round trip, by riding a different ferry on
the return trip?
Honors Advanced Algebra with TrigonometryName: ______
Linear PermutationsDate: ______
Probability Notes (Day 2)Period: ______
Warm-Up:
1. Suppose Moe picks a number from 1 to 4, Larry picks a number from 5 to 7, and Curly picks the
number 8 or 9. How many sets of three numbers are possible? Write the sample space for at least 6
possibilities followed by ”…”.
2. How many different three-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, and 7 if no digit is
used twice?
3. Superstitious Sandy, the coach of a baseball team, starts the same nine players in every game but arranges
them in a different batting order each time. How many games must the team play in order for Sandy to use
all possible batting orders?
4. Goo Lash Caterers provided a seven-course dinner for a party. Unfortunately, they forgot to label the
containers of food, so the courses had to be served in random order. In how many different orders could the
courses have been served?
______
Target Goal: Solve problems using permutations.
In some of the problems above, we multiply a number by every number below it. This is called a FACTORIAL; we use an exclamation mark to signify a factorial.
exs.
To find the factorial of a positive integer by graphing calculator:
1. enter the number you are trying to find the factorial of2. MENU, PRB, 4: !, ENTER
In warm-up problem #1, order is not important. Larry, Moe, and Curly can choose their numbers in any order, that is, they are independent events. In the other four problems, however, order is important. That is, the selection of the second event depends on the first event. When ORDER MATTERS, we call these problems PERMUTATIONS. A linear permutation is the arrangement of objects in a certain linear order.
Definition of P(n, r) – the number of permutations of n objects taken r at a time is defined as follows:
To find the number of permutations of n objects by graphing calculator:
1. enter n number of objects 2. MATH, PRB, 2: nPr 3. enter r number of objects taken at a time
Ex 1: How many ways can 3 books be placed on a shelf if chosen from 8 different books?
Ex 2:From a committee of 18 people, how many ways can a president, vice-president, and treasurer be assigned?
Ex 3: How many different 4-digit garage codes can be made if there are no repeats permitted?
Permutations with repetitions – the number of permutations of n objects of which p are alike and q are alike is:
Ex 4: There are 3 identical white flags and 5 identical blue flags that are used to send signals. All 8 are to be
arranged in a row. How many signals can be given?
Directions:Determine which scenario has repeats and which does not, then choose the appropriate way to solve each.
Ex5: How many ways can the letters from the following words be arranged?
a. FAREWELLb. ILLINIc. COLLECTION
Ex6:.How many ways can 2 different geometry, 4 different geography, 5 different history, and 3 different physics books be arranged on a shelf bysubject?
Ex7:How many ways can 2 geometry, 4 geography, 5 history, and 3 physics books be arranged on a shelf (not by subject) if all books within a subject are indistinguishable?
Ex8: How many ways can the digits from 755,232 be arranged?
Ex9: We have a group of gems to design a bracelet, including 7 emeralds, 4 rubies, and 3 diamonds.
a. How many different bracelet designs are possible if each gem is unique?
b. How many different bracelet designs are possible if each type of gem is indistinguishable?
Ex10:Determine whether the warm-up problems are dependent or independent, then rework the dependent problems in terms of a permutation.
Assignment #2: Worksheet
Honors Advanced Algebra with TrigonometryName: ______
Linear PermutationsDate: ______
Assignment #2Period: ______
Review the basic counting principle by completing the following problems.
1. A box contains 20 muffins, 5 of which are blueberry and the rest of which are raisin. In how many ways can a person choose 2 blueberry muffins and 3 raisin muffins if the first choice must be a raisin muffin and the two kinds must be chosen alternately?
2. The Thirty-Seven Flavors ice cream shop offers 37 different kinds of ice cream. How many different
2-scoop cones can be ordered if customers are given a choice of plain or sugar cones also? (Assume that the customers asking for two different flavors can specify the order in which the flavors are put on the cone.)
Show how the following problems can be completed using (a) the basic counting principle and
(b) permutations.
3. How many different four-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7, and 8 if no
digit is used twice?
a.b.
4. In how many orders can the numbers 11, 22, 33, 55, 88, 99, and 101 be listed?
a.b.
5. How many batting orders can a coach make of 9 players from a roster of 12 players?
a.b.
6.Radio station WOLD plays ten requests between 10:00 and 11:00. In how many different orders can each group of ten songs be played?
a.b.
How many different ways can the letters of each word be arranged?
7.SEE8. LEVEL
9. ALASKA10. PERPENDICULAR
Solve each problem. Be sure to consider repeats when applicable.
11. Don has 5 pennies, 3 nickels, and 4 dimes. The coins of each denomination are indistinguishable.
How many ways can he arrange the coins in a row?
12. How many 6-digit numbers can be made using the digits from 833,284?
13. How many ways can 4 nickels and 5 dimes be distributed among 9 children if each is to receive one coin?
14. Madame Estelle designs jewelry. She is designing a bracelet that will contain a gem in each link of the
bracelet. She has 8 emeralds, 5 rubies, and 3 diamonds. The gem in each type of stone are
indistinguishable from one another. How many different bracelet designs are possible?
Write the sample space for the following problem.
15. Monica, Rachel, and Phoebe like to compete for first, second, and third place in their math class. Write the
sample space for possible lineups.
Honors Advanced Algebra with TrigonometryName: ______
CombinationsDate: ______
Probability Notes (Day 3)Period: ______
Warm-Up:
1. How many different ways can the letters of PARALLEL be arranged?
2. How many different 3-digit numbers can be arranged from the digits 5, 6, 7, 8, or 9…
a. if digits may repeat?b. if digits may not repeat?
3. Which part from #3 represents a permutation? Why? Solve using this idea.
4. From a team of 16, how many different batting orders of 9 can be written?
5. Five mystery and four romance novels are to be arranged on a shelf. How many ways can they be arranged
if all the mystery books must be together?
______
Target Goal: Solve problems involving combinations..
Yesterday, we focused on permutations which were dependent systems where the order was important. In #5 above, for example, the order is important and thus it is a permuatation. What if the problem asked how may different teams of nine can be chosen (instead of “lined up”)? Notice here that the order is not important. This is called a COMBINATION.
Determine whether the following are permutations or combinations:
a. How many committees of three can represent our class?
b. How many ways can we assign a president first, vice president second, etc. to that committee.
c. How many rummy hands of 7 cards exist?
d. Placing 8 books on a shelf.
e. Making a classroom seating chart.
f. Finding the number of diagonals of a polygon.
Definition of C(n,r) – the number of combinations of n distinct objects taken r at a time is defined as
To find the number of combinations of n objects by graphing calculator:
1. enter n number of objects 2. MATH, PRB, 3: nCr3. Enter r number of objects taken at a time
Examples:
1. How many committees of 5 students can be selected from a class of 25?
2. A box contains 12 black and 8 green marbles. How many ways can 3 black and 2 green marbles be chosen?
3. Given 7 distinct points in a plane, how many line segments will be drawn if every pair of points is
connected?
4. How many different 13-card bridge hands can be dealt from a standard deck of 52 cards?
5. From a deck of 52 cards, how many 5 card hands can be selected to meet the following conditions?
a. hand has exactly 3 aces
b. hand has at least 3 aces
Assignment #3: Worksheet
Honors Advanced Algebra with TrigonometryName: ______
CombinationsDate: ______
Assignment #3Period: ______
Determine whether each situation involves a permutation or a combination.
1.a hand of 5 cards from a deck of cards
2.matching the answers on a true/false test
3.putting students in assigned seats
4.a 2-man, 2-woman subcommittee of the Campaign Funds committee that has 8 men and 7 women
Solve each problem.
5.You are required to read 5 books from a list of 12 great American novels. How many different groups can be selected?
6.There are 85 telephones in the editorial department of ‘TEEN magazine. How many 2-way connections can be made among the office phones?
7.There are 27 people in an Algebra class, but only 25 computers in the computer lab, so each student must take turns going to the lab. How many different groups of 25 can the teacher send to the lab?
8.From a deck of 52 cards, how many 7 card hands can be selected with exactly 2 aces?
9.From a deck of 52 cards, how many 7 card hands can be selected with at least 3 aces?
A bag contains 9 blue, 4 red, and 6 white chips. How many ways can 5 chips be selected to meet each
condition?
10.all blue
11.all red
12. 2 red and 3 blue
From a group of 8 juniors and 10 seniors, a committee of 5 is to be formed to discuss plans for the spring
dance. How many committees can be formed given each condition?
13. 3 juniors and 2 seniors
14. all seniors
Honors Advanced Algebra with TrigonometryName: ______
Permutations and CombinationsDate: ______
Assignment #3.5Period: ______
Use the basic counting principle, permutations or combinations to complete the following problems.
1. To create an entry code, you must first choose 3 letters and then 5 single-digit numbers.
- How many entry codes can you create if the letters and digits can repeat?
- How many entry codes can you create if the letters and digits cannot repeat?
2. In arace of twenty people, how manydifferent ways can the first three runners arrive at the finish line?
3. In how many orders can you read nine books out of thirteen in your room?
4.How many groups of four books can be selected from the thirteen in your room?
5. How many ways can the letters in the word LINEAR be arranged?
6. How many ways can the letters in the word PARALLELOGRAM be arranged?
7.Kathy has 4 pennies, 5 nickels, and 6 dimes. The coins of each denomination are indistinguishable. How many ways can she arrange the coins in a row?
8. How many ways can 3identical pens and 4identical pencils be distributed among 7student if each is to receive one writing utensil?
9.How many groups of five students can be selected to represent our class of thirty students?
10.How many ways can the top five students place (highest grade, next highest…5th highest) out of the thirty in our class?
11.A bag contains 8 blue, 7 red, and 10 white chips. How many ways can 6 chips be selected to meet each condition?
a.all blue
b.all red
c. 2 blue and 4 red
12.How many different shirts can you get from a store that has three styles in seven colors and four sizes?
13.A box contains 12 donuts, 4 of which are vanilla covered and 8 of which are chocolate covered. How many groups of 2 vanilla covered and 3 chocolate covered donuts can be chosen from the box?
14.Jessica was asked to choose five paintings from a collection of eight and hang them on the wall in a row. How many different ways could the wall be decorated?