CHAPTER 10: Mathematics of Population Growth



SECTION 15.1 Random Experiments and Sample Space

RANDOM EXPERIMENT:

Any activity whose outcome ____________________ be predicted ahead of time.

Examples:

o Tossing a coin

o Rolling a pair of dice

o Drawing a card out of a deck

o Results of a game

o Forecasting the weather

SAMPLE SPACE: set of all possible outcomes of a random experiment

Notation:

o S = Sample Space

o

o N = SIZE of sample space

Example #1: Toss a coin once and observe whether it lands heads or tails.

▪ Sample Space:

▪

▪ Sample Space Size:

Example #2: Toss a coin twice and observe whether it lands heads or tails for each toss.

▪ Sample Space:

▪

▪ Sample Space Size:

Example #3: A couple plans to have 3 children and considers when they may have a boy or girl.

▪ Sample Space:

▪ Sample Space Size:

Example #4: Rolling a pair of dice simultaneously and consider the TOTAL of the two dice

▪ Sample Space:

▪

▪ Sample Space Size:

Example #5: Sara, Krista, and Arlyn are running for Class President and Vice President.

▪ Sample Space:

▪ Sample Space Size:

TERMINOLOGY DECK OF CARDS: 52 total Cards

• 2 Colors = Red or Black

• 4 Suits = Hearts, Diamonds, Spades, Clubs

• 13 Values = A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

• Face Cards = J, Q, K

Example #6: Pulling a Red Face Card from a standard deck on a single draw.

▪ Sample Space:

▪ Sample Space Size:

Example #7: Top Sellers: 3 boys and 4 girls are doing a class fundraiser of magazines. Consider how the top 2 sellers could be the same gender.

▪ Sample Space:

▪

▪ Sample Space Size:

Example #8: Multiple Choice Test: Consider a multiple choice test with answer options (A, B, C, and D). Consider the possible answer keys that could be made for a 3 question test.

▪ Sample Space:

▪

▪ Sample Space Size:

Example #9: Ranking Candidates: Five candidates (A, B, C, D, and E) are running in an election. The top 3 finishers are chosen as President, VP, and Secretary.

▪ Sample Space:

▪ Sample Space Size:

15.1 HOMEWORK: p. 531 # 1 - 4

15.1 HOMEWORK: p. 531 # 1 – 4

1) Write out the sample space for each of the following experiments.

a. A coin is tossed three times in a row and we observe each toss whether it lands heads or tails.

b. A coin is tossed three times in a row and we observe the number of times it lands tails.

c. A person shoots three consecutive free throws and we observe the number of missed free throws.

2) Write out the sample space for each of the following random experiments.

a. A coin is tossed four times in a row and we observe each toss whether it lands heads or tails.

b. A student randomly guesses the answer to a four question true-false quiz and we observe the student’s answers.

3) Four names (A, B, C, D) are written each on a separate slip of paper, put in a hat, and mixed well. The slips are randomly taken out of the hat, one at a time, and the names recorded.

a. Write out the same space for this random experiment.

b. Find N.

4) A gumball machine has gumballs of four different flavors: apple (A), blueberry (B), cherry (C), and doublemint ( D). The gumballs are well mixed and when you drop a quarter in the machine you get two random gumballs. Write out the sample space for this random experiment.

15.1 HOMEWORK: p. 531 # 1 – 4

1) Write out the sample space for each of the following experiments.

a. A coin is tossed three times in a row and we observe each toss whether it lands heads or tails.

b. A coin is tossed three times in a row and we observe the number of times it lands tails.

c. A person shoots three consecutive free throws and we observe the number of missed free throws.

2) Write out the sample space for each of the following random experiments.

a. A coin is tossed four times in a row and we observe each toss whether it lands heads or tails.

b. A student randomly guesses the answer to a four question true-false quiz and we observe the student’s answers.

3) Four names (A, B, C, D) are written each on a separate slip of paper, put in a hat, and mixed well. The slips are randomly taken out of the hat, one at a time, and the names recorded.

a. Write out the same space for this random experiment.

b. Find N.

4) A gumball machine has gumballs of four different flavors: apple (A), blueberry (B), cherry (C), and doublemint ( D). The gumballs are well mixed and when you drop a quarter in the machine you get two random gumballs. Write out the sample space for this random experiment.

SECTION 15.2 Counting Sample Spaces

COUNTING PROBLEMS:

Find the # of ways…

(1) something can happen

(2) to perform an operation

(3) an event can occur

Examples Questions:

▪ How many ways can you shuffle a deck of cards?

▪ How many NC license plates are there?

(3 letters and 4 numbers)

▪ How many phone numbers are in an area code?

Example #1: A bakery sells 4 different soups at lunch and 7 different sandwiches. Consider the below cases and determine how many different lunches could each person order?

#1a: Michael goes in and plans to order a soup or a salad for his lunch.

#1b: Jarrod goes in and plans to order a soup and a salad for his lunch.

Example #2: A pin code requires four characters to your debit. Consider the below cases and determine how many different pin codes are possible?

#2a: Pin codes uses only numbers.

#2b: Pin code uses only letters.

#2c: Pin code can use only all letters or numbers.

#2d: Pin codes can use any letters or numbers.

TWO BASIC RULES OF COUNTING THEORY

#1: MULTIPLICATION (“And Then”): When something is done in operations (stages or steps), the total number of ways it can be done is found by MULTIPLYING the number of ways/options each operation has.

KEY IDEAS about the MULTIPLICATION RULE:

Many items are being put together to create a larger “whole object”

The concept of ORDER to our operations, stages, or steps is required.

How many numbers should be multiplied together?

What should each one of the numbers you multiply represent?

#2: SUM (“Or”): If one operation can occur in N1 ways and a second operation can occur in N2 (different) ways, then there are exactly N1 + N2 ways in which either the first operation or the second operation can occur (but not both).

Key Ideas of the Sum Rule:

The different groups being added together should not have any outcomes in common.

The sum rule is often used in problems that also require the multiplication rule.

Multiplication #1: Roger has packed 4 pairs of shoes, 6 pants, 7 shirts, and 3 jackets for a week’s vacation to the mountains. How many different outfits could Roger wear if he plans to wear shoes, pants, a shirt, and a jacket?

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#2: A local diner offers a 4 course meal of an appetizer, soup, entrée, and dessert in addition to a drink choice. The menu lists 5 appetizers, 3 soups, 9 entrees, 6 desserts, and 11 drinks. 2a – Multiplication: How many different 4-course meals could be made from this menu?

| | | | | |

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2b – Sum/ Multiplication: How many different meals include an appetizer or dessert?

Example #3: How many ways can you flip a coin 3 times and have a tail as your first flip or a tail as your 3rd flip?

Why does the sum rule not apply to this question?

Example #4: An NFL team has enough room on their roster to sign two more free agents and must make the choice from 3 quarterbacks, 4 linebackers, and 5 wide receivers. How many different ways are there to pick two players if they must play different positions?

15.2 BASIC COUNTING PRACTICE PROBLEMS

#1: Congress consists of 100 senators and 435 representatives.

1a. How many different ways can a delegation be picked if it consists of one senator AND one representative?

1b. How many different ways can a delegation be picked if it consists of one senator OR one representative?

1c. How many different ways can a delegation be picked if it consists of two senators OR two representatives?

#2: A restaurant has 4 soups, 6 salads, and 7 entrees on it’s menu.

2a. How many three course meals (soup, salad, and entrée) are possible?

2b. For lunch the restaurant offers as special of a soup or salad with an entrée. What is the number of possible lunch specials that you could order?

2c. When you show up for lunch, you aren’t really hunger and only want to have a soup, salad, or entrée for your lunch. How many different meals could you eat?

#3: Standard issue NC License plates contain 3 capital letters and 4 numbers.

3a. How many possible NC license plates are there?

|1st letter |2nd letter |3rd letter |1st digit |2nd digit |3rd digit |4th digit |

| | | | | | | |

3b. How many NC license plates have only odd numbers and vowels?

#4: How many ways can you shuffle a deck of cards?

#5a: How many phone numbers are in any area code? Assume number cannot begin with zero

#5b: How many the numbers in an area code start with a 6?

#5c: How many the numbers in an area code start with an even and end with an odd digit?

#6: Five candidates (A, B, C, D, and E) are running in an election. The top 3 finishers are chosen as President, VP, and Secretary.

#7a: How many non-negative integers less than 10,000?

|1000’s Digit |100’s Digit |10’s Digit |1’s digit |Integers |

| | | | | |

#7b: How many non-negative integers less than 10,000 contain only even digits?

#7c: How many non-negative integers less than 10,000 start with 2 prime digits?

#7d: How many non-negative integers less than 10,000 contain no 3 or 5 digits?

#7e: How many non-negative integers less than 10,000 contain any 3 or 5 digits?

#8: Consider all of the numbers from 10,000 – 99,999. (Notice: first digit must be 1 – 9)

8a. How many numbers contain all even digits?

8b. How many numbers contain first and last digits that are odd?

8c. How many numbers cannot contain a repeated digit?

8d. How many numbers contain all of the same number for its digit?

#9: A committee is to be chosen from among 8 scientists, 7 psychics, and 12 clerics. If the committee is to have two members of different backgrounds, how many such committees are there?

#10: Number of different outcomes to flipping a coin 4 times?

#11: How many true or false answer keys are possible for a 4 question test?

#12: A pizza place offers a special on Saturday nights. Starting from a cheese pizza, customers can choose from 4 different toppings (pepperoni, mushroom, sausage, and olives) to make a pizza. How many different pizzas could be made?

#13: A password is 4 characters long. The first character must be a letter and the last number must be a number. How many passwords are possible?

13a. Password IS NOT case sensitive = Uppercase and Lowercase letters are considered the same.

13b. Password IS case sensitive = Uppercase and Lowercase letters are considered different.

#14: A student college ID contains 8 digits to use a meal plan, a 4-digit pin code gains the student access recreational facilities, and an email password contains 6 characters that can be digits or letters (not case sensitive). What is the total number of passwords or IDs that a university computer must be able to hold?

#15: How many 5-letter “words” either start with d or do not have the letter d?

(Note: “words” are any combination of letters with repetition allowed)

15.2 HOMEWORK: pp. 531-532 #9, 10- 18 (even)

15.2 HOMEWORK: pp. 531-532 #9, 10- 18 (even)

9) A California license plate starts with a digit other than 0, followed by 3 capital letters followed by 3 more digits (0 through 9)

a. How many different California license plates are possible?

b. How many California license plates start with a 5 and end with a 9?

c. How many different California license plates have no repeated symbols?

10) A computer password consists of four letters (A through Z) followed by a single digit (0 through 9). Assume that the passwords are not case sensitive (upper and lower case letters treated as the same)

a. How many different passwords are possible?

b. How many different passwords end in 1?

c. How many different passwords do not start with Z?

d. How many different passwords have no Zs in them?

12) A French restaurant offers a menu consisting of three different appetizers, two different soups, four different salads, nine different main courses, and five different desserts.

a. A fixed-price lunch meal consists of a choice of appetizer, salad, and main course. How many different lunches are possible?

b. A fixed-price dinner meal consists of a choice of appetizer, soup or salad, main course, and a dessert. How many different dinners are possible?

c. A dinner special consists of a choice of soup, or salad, or both, and a main course. How many different dinners are possible?

14) Four men and four women line up at a checkout stand in a grocery store.

a. In how many ways can they line up?

b. In how many ways can they line up if the first person in line must be a woman?

c. In how many ways can they line up if they must alternate by gender and a woman must be first?

16) The ski club at Tasmania State University has 35 members (15 females and 20 males). A committee of four members – President (P), Vice President (VP), Treasurer (T), and Secretary (S) – must be chosen.

a. How many different 4-member committees can be chosen?

b. How many different 4-member committees can be chosen if the P and T must be a female?

c. How many different 4-member committees can be chosen if the committee must have 2 females and 2 males?

18) How many 10 digit numbers (ie between 1,000,000,000 and 9,999,999,999)

a. have no repeated digits?

b. are palindromes? (a number that reads the same forward as backwards, 14541)

15.2 HOMEWORK: pp. 531-532 #9, 10- 18 (even)

9) A California license plate starts with a digit other than 0, followed by 3 capital letters followed by 3 more digits (0 through 9)

a. How many different California license plates are possible?

b. How many California license plates start with a 5 and end with a 9?

c. How many different California license plates have no repeated symbols?

10) A computer password consists of four letters (A through Z) followed by a single digit (0 through 9). Assume that the passwords are not case sensitive (upper and lower case letters treated as the same)

a. How many different passwords are possible?

b. How many different passwords end in 1?

c. How many different passwords do not start with Z?

d. How many different passwords have no Zs in them?

12) A French restaurant offers a menu consisting of three different appetizers, two different soups, four different salads, nine different main courses, and five different desserts.

a. A fixed-price lunch meal consists of a choice of appetizer, salad, and main course. How many different lunches are possible?

b. A fixed-price dinner meal consists of a choice of appetizer, soup or salad, main course, and a dessert. How many different dinners are possible?

c. A dinner special consists of a choice of soup, or salad, or both, and a main course. How many different dinners are possible?

14) Four men and four women line up at a checkout stand in a grocery store.

a. In how many ways can they line up?

b. In how many ways can they line up if the first person in line must be a woman?

c. In how many ways can they line up if they must alternate by gender and a woman must be first?

16) The ski club at Tasmania State University has 35 members (15 females and 20 males). A committee of four members – President (P), Vice President (VP), Treasurer (T), and Secretary (S) – must be chosen.

a. How many different 4-member committees can be chosen?

b. How many different 4-member committees can be chosen if the P and T must be a female?

c. How many different 4-member committees can be chosen if the committee must have 2 females and 2 males?

18) How many 10 digit numbers (ie between 1,000,000,000 and 9,999,999,999)

a. have no repeated digits?

b. are palindromes? (a number that reads the same forward as backwards, 14541)

SECTION 15.3 Permutations

WARM UP PROBLEMS:

1) How many six-digit numbers (between 0 and 999,999) have no repeated digits?

2) A 5 character password is not case sensitive. How many passwords use only letters without reusing a letter?

3) 7 people are standing in line at the DMV. How many different ways could these people arrived at the DMV?

4) How many different ways can 8 racers finish 1st, 2nd, and 3rd in the 100 meter dash?

WHAT DO ALL OF THESE COUNTING PROBLEMS HAVE IN COMMON?

PERMUTATIONS (Permute means to “order”)

KEY COMPONENTS:

• ORDER of the objects matters

o Different places or characters in a password, number, line, arrangement

o Different jobs, duties, or positions

• Objects cannot be reused in the arrangement: NO REPLACEMENT or REPETITION

PERMUTATIONS: an _______________________________ arrangement of objects from a group of objects

▪ Notation:[pic]= number of ways to order r objects from n total objects

|Placement |1st |2nd |3rd |… |(r - 1)st |rth |

|# of objects available|n |n – 1 |n – 2 |… |n – (r -1) +1 |n – (r ) + 1 |

| |all items |1 item used |2 items used | |n – r + 2 |n – r + 1 |

MULTIPLICATION RULE:[pic]

▪ Formula:[pic]

Write the warm up problems in permutation notation and check the answer is correct.

Example #1: Use Formula or Multiplication statement to calculate the following values of a permutation and show your work.

a. [pic]=

b. [pic]=

c. [pic] =

d. [pic]=

e. [pic]=

f. [pic]=

Example #2: Consider the letters {a, b, c, d, e, f, g, h} How many 4-letter “words” can be made?

a. Letters can be reused.

b. Letters cannot be repeated.

c. No repetition and d is the last letter.

Example #3: 7 candidates are planning to interview for a job, but there are only 4 interview slots at 1 pm, 2 pm, 3 pm, and 4 pm with the company. How many different interview schedules can the company create?

Example #4: A car’s new stereo system has a 6 slot CD changer (labeled 1 through 6). Jeff has 20 CDs that he regularly listens to. How many ways can Jeff put his CDs into his new car stereo system?

Example #5: A pass code is 8 characters long with no repeated characters, but the first 5 characters have to be lower case letters and the last 3 character must be a number. Let’s see how we can treat this as two permutations with the multiplication rule.

Example #6: 4 people are running for the position of President, Vice President, Treasurer, and Secretary. How many different ways could these people hold those four positions?

Example #7: On the first day of class 30 students find themselves in a classroom with 30 desks already arranged for them, but no seating chart has been made. The teacher gives the students free seating and will write down the seating chart afterwards. How many different seating charts are possible in this situation?

Example #8: Consider the set {A, B, C}. How many different 3-letter words are we allowed to make without repeating a letter in the word?

Why can’t a permutation be used to find the results of 3 consecutive flips of a coin or 4 tosses of a dice?

HOMEWORK: Write with permutation notation and multiplication statement

1) Calculate each expression (a) [pic] (b) [pic] (c) [pic] (d) [pic]

2) The board of directors of a corporation has 12 members. How many ways can one choose a committee of 3-members (President, Vice President, and Secretary)?

3) There are 119 Division 1A college football teams. How many Top 25 rankings are possible?

4) A pin code uses four digits and no digit used can be repeated in the code,

(A) how many different codes are there:

(B) If the first digit cannot be 0?

5) 4 seniors and 3 juniors are waiting in line to buy prom tickets. How many ways can the students stand in line if …

(A) any student can stand anywhere in line?

(B) a senior must be first in the line?

(C) the seniors are the first 4 places?

SECTION 15.3 Subsets and Combinations

COMBINATIONS: Baskin-Robbins and its “31 flavors” of ice cream.

1) How many ways can you get two scoops of different ice cream?

2) How many ways can you get 3 scoops of different ice cream?

Do you see any problem in the way you counted those two outcomes?

Combinations: an ___________________________________________ selection of objects

▪ [pic]= the number of ways to select r objects (items) from n objects (items)

Formula[pic]

Key Components for a Combination:

• Order of the objects DOES NOT matters

▪ Identical Objects

▪ Non-unique items like 4 boys v. saying Mike, Bryan, Eugene, and Karl

▪ Objects cannot be reused in the arrangement: NO REPLACEMENT or REPETITION

Example #1: Calculate the following values use the combination formula.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

MULTIPLICATION RULE V. PERMUTATIONS V. COMBINATIONS

Example 2: Consider the digits {1, 2, 3, 4}

• How many 2-digit numbers can be made from this set?

• How many ways can we select any group of two numbers from {1, 2, 3, 4}?

• How many ways can two numbers be picked in order from {1, 2, 3, 4}?

Example #3: To win the jackpot in a lottery you must select six numbers from 1 to 53. How many possible lottery combinations are there?

a. If you can select the same number as many times as you want and the order mattered?

b. If you cannot select the same number and you win with having the numbers in any order?

c. If you cannot select the same number and you only win with having the numbers in the same order as what is drawn?

Example #4: If there are 7 possible meeting times and a committee must meet 3 times, the number of ways to assign the meeting times is …

Example #5: The number of 5-member delegations that can be created from a 9 person group.

a. How many delegations can be selected?

b. How many delegations can be selected if the members are assigned as a speaker, recorder, researcher, facilitator, and administrator?

Example #6: A necklace of 10 beads can be made from blue, gold, or white beads. How many necklaces have exactly THREE blue, TWO gold, and FIVE white beads?

Example #7: The number of 9 digit codes with exactly FIVE even and FOUR odd digits.

SUBSETS: contains at most ALL the elements or at least NONE of the elements of the given set, and ORDER of the elements is NOT important

Consider the set {a, b, c}. How many subsets are there of this set?

In general, the number of subsets of an n-element set = 2n

▪ YES or NO to each item of n being in a subset = total number of subset

▪ We count the empty set.

SPECIAL PROPERTY between Combinations and Subsets:

Property: The number of subsets of n elements also is a sum of combinations:

[pic]

▪ Each stage represents a subset of a different size 0, 1, 2, 3, … , n which are unordered

Example #8: In a pizza parlor there are 8 different toppings to add to a cheese pizza.

a. How many different pizzas can the parlor make?

b. Exactly how many different 3 topping pizzas could you make?

c. If there were also 3 crust and 4 size options for your pizza how many different 3 topping pizzas could be ordered?

d. The number of pizzas with at most 3 toppings.

e. The number of pizzas with at least 3 toppings.

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Calculator: Permutations

Total # (n) , MATH, PRB, nPr, # of objects (r)

Calculator: Combinations

Total # (n) , MATH, PRB, nCr, # of objects (r)

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