CHAPTER 10: Mathematics of Population Growth



SECTION 15.1 Random Experiments and Sample Space

RANDOM EXPERIMENT: Any activity whose ___________________________________ be predicted ahead of time.

▪ Examples:

SAMPLE SPACE: set of ____________________________ outcomes of a random experiment

NOTATION:

o S =

o N =

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:

BIT STRING: use the numbers 1 and 0 as its digits or “bits” a bit string can only

▪ Length of a bit string = number of digits in the string

▪ Examples:

Example #3: The number of bit strings of length 3?

▪ 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:

Example #6: 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 #7: 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:

Examples #6 and #7 raise the more important question of Counting Theory:

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- Multiplication Rule – Part 1

COUNTING PROBLEMS: Find the # of ways…

(1) something can happen, (2) to perform an operation, or (3) an event can occur

▪ Examples Questions:

▪ How many bit strings of length n?

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

▪ How many bridge hands are possible? (set of 13 cards)

▪ How many NC license plates are there? (3 letters and 4 numbers)

▪ How many phone numbers are in an area code?

MULTIPLICATION RULE: 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.

The total number of ways of performing OP.1, then OP.2, then OP.3, then …, OP.k is

N1 * N2 * N3 * … * Nk (ORDER MATTERS)

▪ Operations are being put altogether to create a whole

▪ KEY TERM: “AND, AND THEN” statements

15.2 MULTIPLICATION RULE EXAMPLES

#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?

|Shoes |Pants |Shirts |Jackets |

| | | | |

Example #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. How many different 4-course meals could be made from this menu?

|Appetizers |Soups |Entrees |Desserts |Drinks |

| | | | | |

#3a: How many NC license plates are there? (3 letters and 4 numbers)

| | | | | | | |

| | | | | | | |

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

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

| | | | | | | |

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

|1st Card |2nd Card |3rd Card |4th Card |… |2nd to Last |Last |

| | | | | | | |

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

| | | | | | | |

| | | | | | | |

#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?

|Thousand’s Digit |Hundred’s Digit |Ten’s Digit |One’s digit |

| | | | |

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

#8a: Number Bit strings of length 4?

| | | | |

| | | | |

#8b: How many true or false answer keys are possible for a 4 question test?

| | | | |

| | | | |

#8c: 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?

| | | | |

| | | | |

KEY IDEAS about the MULTIPLICATION RULE:

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

#2: Phrase: “AND THEN” = references order

#3: Be careful if repetitions are allowed or not

CLASS WORK PROBLEMS:

#1: A library has 5000 books and the librarians want to encode each using a code word consisting of 3 letters followed by 3 numbers. Are there enough codewords to encode all 5000 books with a unique codeword?

#2: How many m by n matrices are there each of whose entries is 0 or 1?

#3: A musical band has to have at least one member. It can contain at most one drummer, one pianist, one bassist, one lead singer, and at most 2 background singer. How many total bands are there if we consider any two bands the same if they have the same number of members of each category?

#4: How many numbers less than 1 million contain the digit 2?

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

9) A California license plate starts with a 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 15.2 Counting Sample Spaces – Sum Rule - Part 2

Warm Up:

1) 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?

a) Assume the password IS NOT case sensitive.

b) Assume the password IS case sensitive.

2) Using Opposites (Conditionals) to find what you want:

How many non-negative numbers less than 1,000,000 contain 3 or 5?

SUM RULE: 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 TERM: “OR”

▪ GENERAL SUM RULE: For K operations(steps, stages) and Ni = different ways/ options for the ith operation, then

Total Number of Ways for Exactly One Outcome

= N1 + N2 + N3 + … + Nk

▪ WATCH OVERLAP BETWEEN OPERATIONS TO AVOID OVERCOUNTING

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

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

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

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

EXP #2: How many bit strings of length 2 or 3?

EXP #3: How many ways can three digit numbers (100-999) end in a 6 or 9?

EXP #4: How many bit strings of length 3 with 1 in the 1st position or 1 in the 3rd position?

EXP #5: An NFL team has two first round draft picks to make has limited the choice to 3 quarterbacks, 4 linebackers, and 5 wide receivers. How many different ways are there to pick two players if they must play different positions?

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

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

b. 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?

PRACTICE PROBLEMS: Problems use the sum rule, multiplication rule, or both.

#1: 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?

#2: How many numbers are there which have five digits, each being a number in {1, 2, 3, …, 9}, and either having all digits odd or all digits even?

#3: How many 5-letter “words” either start with d or do not have the letter d? (Note: A “word” is any combination of letters with repetition allowed)

#4: Suppose that a pipeline network is to have 30 links. For each link, the pipe’s size may be any one of 7 sizes and made from any one of 3 materials. How many different pipeline networks are there?

#5: 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?

#6: Consider all of the numbers from 10,000 – 99,999.

a. How many numbers contain all even digits?

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

c. How many numbers cannot contain a repeated digit?

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

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?

SPECIAL CASE OF THE MULTIPLICATION RULE

PERMUTATIONS (Permute means to “order” items)

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| | | |… | | |

MULTIPLICATION RULE:

o Product of all the numbers starting at N and counting down to have r total numbers

Formula: [pic]

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

Example #1: 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:

1) Calculate (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) If a telephone extension has four digits, how many different extensions are there with no repeated digits: (A) If the first digit cannot be 0?(B) If the first digit cannot be 0 and the second cannot be 1?

5) 4 seniors and 3 juniors are waiting in line to buy prom tickets. How many ways can the students stand in line if the seniors are the first 4 places and care about where they stand in line?

SECTION 15.3 Subsets and Combinations

WARM UP PROBLEMS:

1) How many 5 character passwords are possible if you are allowed numbers and letters, and it is case sensitive?

a. no restrictions.

b. no repeated characters.

2) A typical combination lock has 40 numbers (0 – 39) and opens by turning clockwise, counterclockwise, and then clockwise. How many different locks can a company manufacture?

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?

COMBINATION: an ________________________ selection of objects

▪ Key Components for a Combination:

o Order of the objects DOES NOT matters

▪ Identical Objects

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

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

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

Formula: [pic][pic]

Example #1: Calculate the following values and show your work

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

MULTIPLICATION V. PERMUTATIONS V. COMBINATIONS

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

• How many 2-digit numbers can be made from this list of digits?

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

• How many ways can two different 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 through 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: The number of delegations to the president consisting of 2 senators and 2 representatives remembering that there are 100 senators and 435 representatives in Congress.

Example #7: The number of bit strings of length 9 with exactly FIVE 1’s and FOUR 0’s.

SUBSETS: contain 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?

List =

Counting =

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 = all NOs.

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.

SECTION 15.4 PROBABILITY SPACES

EVENT: any subset of the sample space or any set of individual outcomes

▪ Simple Event: an event of exactly one outcome

▪ Impossible Event: an event with no outcomes;

▪ Certain Event: an event with all outcomes of the sample space;

EXAMPLE #1: Consider the random experiment of tossing a coin three times.

|SAMPLE SPACE: |

|E1: Toss 2 or more heads. | |

|E2: Toss more than 2 heads. | |

|E3: Toss 2 or less heads. | |

|E4: Toss no heads. | |

|E5: Toss Exactly 1 tail. | |

|E6: First Toss is heads. | |

|E7: Same number of heads as tails | |

|E8: Toss 3 heads or Less | |

EXAMPLE #2: Consider the random experiment of tossing a coin and rolling a die.

|SAMPLE SPACE: |

|E1: An even number. | |

|E2: A head. | |

|E3: A head and an odd number. | |

|E4: A tail and prime number. | |

|E5: A number less than 7. | |

|E6: A number less than 3 | |

|E7: A 7 is rolled | |

PROBABILITY ASSIGNMENT:

Probability Assignment: a function that assigns to each event E a number between 0 and 1, which represents the probability of the event E

▪ Notation: Pr(E)

0 = probability assignment of an impossible event (empty set)

1 = probability assignment of the entire sample space (certain event)

Probability Assignment Requirements:

1. All probabilities are numbers between 0 and 1

2. The sum of the probabilities of the simple events equals 1

EXAMPLE: There are six players in a tennis tournament: Ana (Russian, Female), Ivan (Croatian, male), Lleyton (Aussie, male), Roger (Swiss, male), Serena (American, female), and Venus (American, female).

Sample space is who will win the tournament:

Professional Odds-Maker comes up with the following probability assignment:

Pr(Ana) = 0.08 , Pr(Ivan) = 0.16, Pr(Lleyton) = 0.20, Pr(Roger) = 0.25, Pr(Venus) = 0.16

What is the probability of Serena winning?

Probability represents the probabilities of simple events, and all other probabilities follow by addition.

1) Probability an American will win:

2) Probability a male will win:

3) Probability an American male will win:

4) Probability a European wins:

5) Probability a female will win:

PROBABILITY SPACE: the combination of the sample space and its probability assignment

▪ Sample Space: S = {σ1, σ2, …, σN }

o σ1, σ2, …, σN represents the simple events of the space

▪ Probability Assignment: Pr(σ1), Pr(σ2), …, Pr(σN)

o Each of these numbers is between 0 and 1.

o Pr(σ1) + Pr(σ2) + …+ Pr(σN) = 1

▪ Events: These are all the subsets of S, including {} and S. The probability of an event is given by the sum of the probabilities of the individual outcomes that make up the event.

o Pr({}) = 0

o Pr(S) = 1

EXAMPLE Probability Space: Consider the sample space S = {σ1, σ2, σ3, σ4}.

a. If all outcomes are equally likely, then what are their probabilities?

b. If Pr(σ1) = 0.3, Pr(σ2) = 0.25, and Pr(σ3) = .17, then what is Pr(σ4)?

c. If Pr(σ1) = 0.18, Pr(σ2) = 0.12, and Pr(σ3) = Pr(σ4), then what is Pr(σ4)?

d. If Pr(σ1) = 0.18, Pr(σ2) = 0.22, and Pr(σ3) is double Pr(σ4), then what is Pr(σ4)?

e. If Pr(σ1) is double Pr(σ2) and Pr(σ2) = Pr(σ3) = Pr(σ4), then what are all probabilities?

HOMEWORK

15. 4 EVENTS: p. 534 #42, 44 Write out all events described below in set notation .

42. Consider the random experiment where a student takes a four-question true(T)-false(F) quiz.

a. E1: “exactly 2 of the answers given are Ts.”

b. E2: “at least 2 of the answers given are Ts.”

c. E3: “at most 2 of the answers given are Ts.”

d. E4: “first 2 answers given are Ts”

44. Consider the random experiment of drawing 1 card out of an ordinary deck of 52 cards.

a. E1: “the card drawn is the queen of hearts”

b. E2: “the card drawn is a queen”

c. E3: “the card drawn is a heart”

d. E4: “the card drawn is a face card”

15.4 PROBABILITY ASSIGNMENT: p. 533 #35 – 39

35. Consider the sample space S = {σ1, σ2, σ3, σ4, σ5}. You are given Pr(σ1) = 0.22 and Pr(σ2) = 0.24.

a. If σ3, σ4, and σ5 all have the same probability, find Pr(σ3).

b. If σ3 has the same probability as σ4 and σ5 combined, find Pr(σ3).

c. If σ3 has the same probability as σ4 and σ5 combined, and if Pr(σ5) = 0.1, give the probability assignment for this probability space.

36. Consider the sample space S = {σ1, σ2, σ3, σ4}. Suppose you are given Pr(σ1) + Pr(σ2) = Pr(σ3) + Pr(σ4)

a. If Pr(σ1) = 0.15, find Pr(σ2).

b. If Pr(σ1) = 0.15 and Pr(σ3) = 0.22, give the probability assignment for this probability space.

37. Seven players are entered in a tennis tournament. According to an expert handicapper, P2, P3, …, P7 have the same probability of winning, and p1 is twice as likely to win as one of the other players. Write down the sample space, and find the probability assignment for the probability space defined by this handicapper’s opinion.

38. Six players are entrees in a chess tournament. According to an expert handicapper, P1 has a probability of 0.25 of winning, P2 has a probability of 0.15 of winning, P3 has a probability of 0.09 of winning, and P4, P5, and P6 all have an equal probability of winning. Write down the sample space, and find the probability assignment for the probability space defined by this handicapper’s opinion.

39. A circular spinner has a RED sector with a central angle of 1080, BLUE and WHITE sectors both with central angles of 720, and GREEN and YELLOW sectors both with central angles of 540. Assume that the needle is spun so that it randomly stops at one of these sectors. Describe the sample space for this game and Give the probability assignment for this probability space.

SECTION 15.5 EQUIPROBABLE SPACES PART 1

EQUIPROBABLE SPACES: A probability space where each simple event has an equal probability of occurring (All outcomes are equally likely)

Probability of EVENT, E: The probability of an event E occurring is ratio of the size of event to the size of the sample space.

▪ [pic]

Example #1: 12 boys and 15 girls in class. Probability a boy is called out of class is

Example #2: A die being tossed once.

a. What is the sample space, S?

b. What is N(S)?

c. What is the probability of rolling an even number?

d. What is the probability of rolling a 1?

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

COMPLEMENT: a negation of the original statement = NOT EVENT

1) Event = Pick a Black Card

2) Event = Pick a Heart

3) Event = Flip a Coin twice and get one tail

4) Event = Flip 3 times and get all heads

Probability of COMPLEMENT of Event, Ec:

▪ Example 1: 12 boys and 15 girls in class. Probability a boy is not called out of class is

▪ Example 2: The eye color of students is recorded as 150 blue, 285 brown, 93 green, and 72 hazel. What is the probability that a student doesn’t have blue eyes?

▪ Example 3: If a number from 0 to 999 is to be called out loud. What is the probability that the number will contain at least one 3?

E and EC are called COMPLEMENTARY EVENTS .

▪ The probabilities of complementary events add up to 1: Pr(E) + Pr(EC) = 1

▪ Reminder: Sometimes it’s easier to calculate Pr(EC) instead of Pr(E).

USE COUNTING THEORY TO FIND PROBABILITIES

Example #1: Consider random experiment of tossing a fair coin four times.

a. What is N(S)?

b. How many ways roll exactly one head?

c. What is the probability of rolling exactly one head?

d. What is the probability of rolling at least one head?

Example #2: Bit strings of length 6 chosen at random.

a. How many total bit strings of length 6 are possible?

b. Probability bit string has exactly 3 1’s.

c. Probability bit string has at most 2 1’s.

d. Probability bit string has at least 3 1’s.

Example #3: A couple is planning to have 3 children and is concerned about the gender of their children.

a. How many different ways could they have boys or girls?

b. What is the probability they will have one boy?

Example #3 CONTINUED…

c. What is the probability they will have two girls?

d. What is the probability they will have at most one boy?

e. What is the probability they will have at least one girl?

Example #4: Draw 2 card from a standard deck of 52, putting the card back in the deck before redrawing. (REPLACEMENT IS ALLOWED)

a. How many different ways can two cards be drawn?

b. What is the probability to draw 2 kings?

c. What is the probability to draw 2 face cards?

d. What is the probability to draw a red then a black card?

e. What is the probability to draw a red and a black card?

f. What is the probability to draw a two diamonds card?

g. What is the probability to draw a two of same suit?

SECTION 15.5 EQUIPROBABLE SPACES PART 2

INDEPENDENT EVENTS: two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other.

Example of Independent Events:

#1. Tossing a coin twice times

#2. Rolling a die consecutive times

#3. Choosing a card from a deck, replacing it, and drawing again

#4. 3 red, 6 green, and 5 yellow marbles are chosen from a bag and a marble is replaced after it was drawn.

(Key Concept = REUSING SOMETHING/ REPLACEMENT)

The Multiplication Principle for Independent Events

If events E and F are independent, then the probability that both occur (E and then F) is the product of the respective probabilities.

▪ [pic]

Comparison of Independent and Dependent Multiplication:

|DEPENDENT |INDEPENDENT |

|A deck of cards is shuffled and two cards are dealt. What is the chance that both|A deck of cards is shuffled, a card is drawn, and then replaced in the deck, |

|are aces? |shuffled, and drawn again. What is the chance that both are aces? |

|Probability first is an ace = |Probability first is an ace = |

| | |

|Probability second is an ace = |Probability second is an ace = |

| | |

|Probability (1st and 2nd) = |Probability (1st and 2nd) = |

Example #1: 7 red marbles, 5 green marbles, and 8 yellow marbles are in a bag and each time a marble is chosen it is replaced back in the bag for the next draw.

a. Suppose the first draw is a red. What is the chance of getting a yellow on the second draw?

b. Suppose the first draw is a red. What is the chance of getting a red on the second draw?

Example 1 Continued: 7 red marbles, 5 green marbles, and 8 yellow marbles are in a bag and each time a marble is chosen it is replaced back in the bag for the next draw.

c. Find Pr(Red then Yellow)

d. Find Pr(Yellow then Red)

e. Find Pr(Red and Yellow)

f. Find Pr(Red then Red)

g. Find Pr(Red then Green)

h. Find Pr(Green then Yellow)

Example #3: 7 red marbles, 5 green marbles, and 8 yellow marbles are in a bag and drawn at random with no replacement.

a. Suppose the first draw is a red. What is the chance of getting a yellow on the second draw?

b. Suppose the first draw is a red. What is the chance of getting a red on the second draw?

c. Find Pr(Red then Yellow)

d. Find Pr(Yellow then Red)

e. Find Pr(Red and Yellow)

f. Find Pr(Red then Red)

Example #4 : Roll a die 4 times. What is the probability of getting exactly 3 1’s to show up?

BINOMIAL FORMULA:

Probability that an independent event will occur exactly k times out of n tries =

[pic]

▪ n = number of total trials (tosses of coin, roll of dice, etc)

▪ k = EXACT number of times your outcome will occur

▪ p = probability event will occur at once (single event)

▪ 1 – p = probability event will not occur at one trial (COMPLEMENT)

▪ [pic]= choosing k of the n trials to be the event you want (ways it can occur in all)

Ex #5a: Roll a die 6 times. What is the probability of getting exactly 3 1’s to show up?

Ex #5b: Toss a coin 4 times. What is the probability of getting exactly 3 H’s to show up?

Ex #6: Draws are made at random with replacement from the box containing 8 identical balls marked with {1, 1, 2, 3, 3, 3, 4, 5}.

a. Probability of exactly 20 1’s after 25 draws.

b. Probability of exactly 8 3’s after 16 draws.

c. Probability of exactly 5 5’s after 15 draws.

Equiprobable Spaces Homework: pp. 534 – 535 #47 – 57

47) Consider the random experiment of tossing an honest coin 3 times in a row. Find the probability. (Hint: See Exercise 41)

a. E1: “Tossing exactly 2 heads”

b. E2: “all tosses come out the same”

c. E3: “half of the tosses are heads and half are tails”

d. E4: “first two tosses are tails”

48) Consider the random experiment where a student takes a 4-question true or false quiz. Assume now that the student randomly guesses the answer for each question. Find the probability. (Hint: See Exercise 42)

a. E1: “exactly 2 of the answers given are Ts.”

b. E2: “at least 2 of the answers given are Ts.”

c. E3: “at most 2 of the answers given are Ts.”

d. E4: “first 2 answers given are Ts”

49) Consider the random experiment of rolling a pair of honest dice. Let T2, T3, …, T12 represent the events of rolling “a total of 2”, “a total of 3”, .., “a total of 12”. (Hint: See Exercise 43)

a. Find Pr(T6) and Pr(T8)

b. Find Pr(T5) and Pr(T9)

c. E1: Roll two of a kind; Find Pr(E1)

d. E2: Roll a total of 3 or less; Find Pr(E2)

e. E3: Roll a total of 7 or 11; Find Pr(E3)

50) Consider the random experiment of drawing 1 card out of an honest deck of 52 cards. Find the probability.

a. E1: “the card drawn is the queen of hearts”

b. E2: “the card drawn is a queen”

c. E3: “the card drawn is a heart”

d. E4: “the card drawn is a face card”

51) Consider the random experiment of tossing an honest coin 10 times in a row. Find the probability. (Hint: See Exercises 5 and 45)

a. E1: “toss no tails”

b. E2: “toss exactly 1 tail”

c. E3: “toss exactly twice as many heads as tails”

52) Consider the random experiment of drawing two cards out of an honest deck of 52 cards. Find the probability. The order of the cards does not matter.

a. E1: “draw a pair of queen”

b. E2: “draw a pair”

53) If a pair of honest dice are rolled once, find the probability of

a. rolling a total of 8

b. not rolling a total of 8

c. rolling a total of 8 or 9

d. rolling a total of 8 or more

54) A gumball machine has gumballs of four different flavors: A, B, C, and D. When a 50-cent piece is put into the machine, five random gumballs come out. Find the probability that

a. each gumball is a different flavor

b. at least two of the gumballs are the same flavor.

55) A student takes a 10-question true or false quiz and randomly guesses the answer to each question. Suppose a correct answer is worth 1 point, an incorrect answer is worth -0.5 points. Find the probability that the student

a. gets 10 points.

b. gets -5 points.

c. gets 8.5 points.

d. gets 8 or more points.

e. gets 5 points.

f. gets 7 or more points.

56) Ten names (A, B, C, D, E, F, G, H, I, J) are written on separate slips of paper, put in a hat, and mixed well. Four names are randomly taken out of the hat, one at a time. Assume that the order in which the names are drawn matters. Find the probability that

a. A is the first name chosen

b. A is one of the four names chosen

c. A is not one of the four names chosen

d. the four names chosen are A, B, C, D in that order

57) A club has 15 members. A delegation of four members must be chosen to represent the club at a convention. All delegates are equal, so the order in which they are chosen doesn’t matter. Assume that the delegation is chosen randomly by drawing the names out of a hat. Find the probability that

a. Alice (a club member) is selected

b. Alice is not selected.

c. club members Alice, Bert, Cathy, and Dale are selected

15.6 ODDS

ODDS: For an arbitrary event, odds represent a comparison of the number of ways than even can occur (favorable outcomes) to the number of ways an event does not occur (unfavorable outcomes).

▪ Favorable Outcomes = N(E)

▪ Unfavorable Outcomes = N(Ec)

The odds of (odds in favor of) the event E are given by the ratio N(E) to N(Ec).

The odds against the event E are given by the ratio N(Ec) to N(E).

Example #1 - Odds: Suppose you are playing a game in which you roll a pair of honest dice. If you roll a total of 7 or 11, you win the game. What are your odds of winning in this game? What are your odds of losing?

Win: Odds in Favor of Lose: Odds against

Example #2– Connecting Probability and Odds:

#2a: Steve Nash shoots free throws with a probability of 0.90. For every 100 free throws, Nash will make 90 and will miss 10.

The odds of Nash making a free throw are

The odds against Nash making a free throw

#2b: Shaquille O’Neal shoots free throws with a probability of 0.52. For every 100 free throws, Shaq will make 0.52 and will miss 48.

The odds of Shaq making a free throw are

The odds against Shaq making a free throw are

Example #3: There are six players in a tennis tournament: Ana, Ivan, Lleyton, Roger, Serena, and Venus.The probability assignment for the tournament as follows:

Pr(Ana) = 0.08 , Pr(Ivan) = 0.16, Pr(Lleyton) = 0.20, Pr(Roger) = 0.25, Pr(Serena) = 0.15, Pr(Venus) = 0.16

Express each of these probabilities as odds of winning the tourney and odds against winning the tourney.

|Probability (Find a reduced fraction for the probability) |Odds For |Odds Against |

|Pr(Ana) = 0.08 | | |

|Pr(Ivan) = 0.16 | | |

|Pr(Lleyton) = 0.20 | | |

|Pr(Roger) = 0.25 | | |

|Pr(Serena) = 0.15 | | |

|Pr(Serena) = 0.16 | | |

Example #4: Find the probability of event E for each of the given odds.

d. The odds of event E are 7 to 8

e. The odds of event E are 2 to 15

f. The odds against event E are 13 to 7.

g. The odds against event E are 7 to 3.

h. The odds of event E are 9 to 11.

i. The odds against event E are 1 to 1.

j. The odds of E are the same as the odds against E.

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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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