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:

  • S =
  • 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.

  1. A coin is tossed three times in a row and we observe each toss whether it lands heads or tails.
  2. A coin is tossed three times in a row and we observe the number of times it lands tails.
  3. 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.

  1. A coin is tossed four times in a row and we observe each toss whether it lands heads or tails.
  2. 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.

  1. Write out the same space for this random experiment.
  2. 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 drinkchoice. 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 TasmaniaStateUniversity 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.

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

  1. How many three course meals (soup, salad, and entrée) are possible?
  1. 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.

  1. How many numbers contain all even digits?
  1. How many numbers contain first and last digits that are odd?
  1. How many numbers cannot contain a repeated digit?
  1. 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
  • Different places or characters in a password, number, line, arrangement
  • 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:= number of ways to order r objects from n total objects

Placement / 1st / 2nd / 3rd / … / (r - 1)st / rth
# of objects available / …

MULTIPLICATION RULE:

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

Formula:

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.

b.

c.

d.

e.

f.

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

  1. Letters can be reused.
  1. Letters cannot be repeated.
  2. 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) (b) (c) (d)

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:
  • Order of the objects DOES NOTmatters
  • 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
  • = the number of ways to select or choose r objects (items) from n total objects (items)

Formula:

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

a.

b.

c.

d.

e.

f.

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?

  1. If you can select the same number as many times as you want and the order mattered?
  1. If you cannot select the same number and you win with having the numbers in any order?
  1. 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.