PRE-CALCULUS PROBABILITY UNIT Permutations and Combinations

Fundamental Counting Principal – A way to find the total possible ______________ something can be arranged. 𝑻𝒉𝒆 𝒍𝒖𝒏𝒄𝒉 𝒔𝒑𝒆𝒄𝒊𝒂𝒍 𝒂𝒕 𝒕𝒉𝒆 𝒍𝒐𝒄𝒂𝒍 “𝑮𝒓𝒆𝒂𝒔𝒚 𝑺𝒑𝒐𝒐𝒏” 𝒅𝒊𝒏𝒆𝒓 𝒐𝒇𝒇𝒆𝒓𝒔 𝟑 𝒆𝒏𝒕𝒓𝒆𝒆𝒔, 𝟔 𝒔𝒊𝒅𝒆 𝒅𝒊𝒔𝒉𝒆𝒔 𝒂𝒏𝒅 𝟒 𝒅𝒓𝒊𝒏𝒌𝒔. 𝑯𝒐𝒘 𝒎𝒂𝒏𝒚 𝒎𝒆𝒂𝒍 𝒄𝒉𝒐𝒊𝒄𝒆𝒔 𝒂𝒓𝒆 𝒕𝒉𝒆𝒓𝒆?

A Michigan license plate consists of 3 letters (excluding O and I) followed by 4 numbers. How many different license plates are possible?

PERMUTATIONS A ___________________________________ is a selection of a group of objects in which ________________ is important. A ____________________________________ (___) is a product of a number and the natural numbers that are one less. 0! = ____ 2 Ways To Do a Permutation – A group 8 of marathoners in the Olympics are vying for gold, silver and bronze. How many ways can the medals be awarded?

Use the FCP –

Use the formula - 𝑷𝒓 = 𝒏!

(𝒏−𝒓)!

How many ways can a 4-digit number be formed by using only the digits 5-9 if each digit can only be

used once?

COMBINATIONS A ______________________________ is a grouping of items in which _________________ does not matter. Generally, there are _____________________ ways to select items when order does not matter. How many ways can the letters A, B, and C be arranged if order matters. How many combinations of the letters A, B and C be arranged.

The formula: 𝑪𝒓 = 𝒏!

𝒓!(𝒏−𝒓)!

Farmer Pete is going to butcher 3 chickens for Sunday dinner. How many groups of 3 chickens can Farmer Pete butcher out of his flock of 11?

Homework for Permutations and Combinations 1. An Internet code consists of one digit followed by one letter. The number zero and the letter O are excluded. How many codes are possible? 2. How many distinct songlists can you listen to 4 songs on a playlist of 12 songs? 3. How many arrangements can be made when determining the order in which each class (Freshmen, Sophomores, Juniors and Seniors) finished in the Homecoming Float building competition? 4. A teacher will only allow 3 students to the library at a time. How many groups of 3 students can he choose out of his 25 students? 5. Pizza Larry’s offers 4 different crusts, 15 different toppings and 3 different sizes. How many ways can Pizza Larry serve up a 1 topping pizza? How many different 2- topping pizzas can he serve up?

How many different 3-topping pizzas can he serve up? Compare the following

6. 𝑷𝟑𝟕 ____𝑪𝟒

𝟕 7. 𝑷𝟒𝟕 ____𝑷𝟑

𝟕 8. 𝑪𝟑𝟕 ____𝑪𝟒

𝟕 9. 𝑷𝟏𝟎𝟏𝟎 ____𝑪𝟏𝟎

𝟏𝟎 7. Explore and complete the following pattern:

𝑪𝟎𝟎 ___

𝑪𝟏𝟏 𝑪𝟏

𝟏 ___ ___

𝑪𝟎𝟐 𝑪𝟏

𝟐 𝑪𝟐𝟐 ___ ___ ___

𝑪𝟎𝟑 𝑪𝟏

𝟑 𝑪𝟐𝟑 𝑪𝟑

𝟑 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ What did you discover? Now, expand the Binomial (𝒙 + 𝒚)𝟒

PRE-CALCULUS PROBABILITY UNIT Probability

Probability is the measure of how likely an event can occur.

Theoretical Probability: 𝑷(𝒆𝒗𝒆𝒏𝒕) = # 𝒐𝒇 𝒂 𝒇𝒂𝒗𝒐𝒓𝒂𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆

# 𝒐𝒇 𝒕𝒐𝒕𝒂𝒍 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒐𝒖𝒕𝒄𝒐𝒎𝒆𝒔

Example: A song list has 9 dance songs and 6 slow songs on it. What is the probability that a randomly selected song will be a slow one? Rolling Dice – Two dice are rolled. What is the probability: a. The sum is 10? b. The sum is 7? c. The difference is 6?

1 2 3 4 5 6

1

2

3

4

5

6

Probability of a Complement If the probability of an event E is P(___), then the probability of its ____________________ (not E) is: P(not E) = 1 - _____. Example: The game Battleship is played with 5 ships on a 100-hole grid. Using the chart below, calculate the probabilities of the first guess:

Game Piece Holes

Destroyer 2

Cruiser 3

Submarine 3

Battleship 4

Carrier 5

a. P(Destroyer) b. P(Submarine or Battleship) c. P(miss) Finding Probabilities Using Permutations and Combinations Example: Book-Nerd Billy received his 4-digit library code to use with the library computer. No digit repeats. Billy received the code: 7654. What was the probability that he would receive a code with consecutive digits? Decide: Permutation or Combination Find total outcomes of 4 digits from 10. Find the favorable outcomes: 0123, 1234, . . P(consecutive numbers)

So far, we have calculating Theoretical Probability, but, what if we actually do an experiment.

𝑬𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒏𝒕𝒂𝒍 𝑷𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 = # 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒂𝒏 𝒆𝒗𝒆𝒏𝒕 𝒐𝒄𝒄𝒖𝒓𝒔

# 𝒐𝒇 𝒕𝒓𝒊𝒂𝒍𝒔

Example: A card is chosen from a deck and then replaced. The following table shows the results:

Suit Hearts Diamonds Clubs Spades

Number 5 9 7 5

a. Find the experimental probability of choosing a diamond. b. Find the experimental probability of choosing a card that is not a club. Geometric Probabilities: Find the probability of choosing a point at random inside the shaded area of this equilateral triangle.

14 in

in.

4 in

inin

14 in

in.

4 in

inin

Homework for Probabilities

1. What is the probability of randomly selecting a date during the year that is not in either January or December? 2. There are 12 marbles in a bag: 3 each of blue, green, red and yellow. Three marbles are chosen at random. Find the probability that all 3 of the balloons are green. 3. A clerk has 4 different letters that need to go in 4 different envelopes. What is the probability that all 4 letters are placed in the correct envelopes? 4. What is the probability that a dart will hit in the shaded area? In the bullseye? 5. A cross-country coach must pick 3 runners to run from a team of 8 to run in a big meet. What is the probability that he will choose the 3 strongest runners? 6. A radio station in Michigan is giving away a trip to anywhere in the United States that the winner wants to go. What is the probability that the winner will select a destination that is not a state that borders Michigan? 7. From the table below, select the experimental probability of spinning:

Color Red Green Blue

Spins 5 8 7

a. A red b. Red or Blue c. Not Blue 8. Explain whether the experimental probability of tossing tails when a coin is tossed 25 times is always, sometimes, or never equal to the theoretical probability. Why?

2

2

4

PRE-CALCULUS PROBABILITY UNIT Independent vs. Dependent Events

Events are considered __________________________________ if the occurrence of one event does ________ affect the probability of another.

Probability of Independent Events 𝑷(𝑨 𝒂𝒏𝒅 𝑩) = 𝑷(𝑨) × 𝑷(𝑩)

What different activities can you think of where the events are always independent of each other? What is the probability of spinning a 3 and then a 5? What is the probability of spinning an even number and then a 7?

Probability of Dependent Events

Events are considered _________________________________ if the occurrence of one event _________________ the probability of another.

𝑷(𝑨 𝒂𝒏𝒅 𝑩) = 𝑷(𝑨) × 𝑷(𝑨 𝑩)⁄ Two cards are drawn from a standard deck. Find the probabilities that: a. An ace is drawn, then replaced and another ace is drawn. 𝑷(𝑨𝒄𝒆 𝑨𝒄𝒆⁄ ) = b. A spade is drawn and then a Heart 𝑷(𝑺𝒑𝒂𝒅𝒆 𝑯𝒆𝒂𝒓𝒕) = ⁄

Homework for Independent and Dependent Events 1. Find the probabilities: a. Rolling a 1 and then another 1 on a dice. b. Having a coin land as heads three time in a row. 2. The table shows a quality control study from a lightbulb factory.

Shipped Not Shipped

Defective 10 45

Not Defective 942 3

a. Find the probability that a shipped bulb is not defective

b. Find the probability that a bulb is defective and shipped. 3. The table shows the employment (in millions) by education level for adults ages 21-24.

Education Level Employed (millions) Not Employed (millions)

No High School Diploma 1.060 0.834

High School Diploma 2.793 1.157

Some College 4.172 1.634

Bachelor’s Degree 1.53 0.372

Advanced Degree 0.104 0.041

a. Find the probability that a person with an advanced degree is employed. b. Find the probability that person without a high school graduate is not employed. 4. Determine whether the events are dependent or independent. a. A coin comes up heads and a dice rolled at the same time comes up a 6. b. A 4 is drawn from a deck of cards, falls to the floor , and then an ace is drawn. c. A 1 is rolled on a dice and then a 4 is rolled on the same dice. d. A dart hits a bull’s eye and a second dart also hits the bull’s eye. 5. True or False – If a coin is tossed 10 times and it comes up heads each time, there is a greater probability that a tails will come up on the 11th toss. 6. A bag contains 20 checkers – 10 red and 10 black. a. What is the probability that a black is drawn, replaced and another black is drawn? b. What is the probability that a black is drawn, not replaced, then another black is drawn?

PRE-CALCULUS PROBABILITY UNIT Binomial Distributions

Recall Pascal’s Triangle: Converting to Combinations: Binomial Theorem –A way of expanding binomials instead of using __________________________ instead of Pascal’s Triangle. Expand: (𝟑𝒑 + 𝒒)𝟑 Binomial Experiment: An experiment where the outcome has an equal chance of being a ____________________ or a _____________________. Binomial Probability: 𝑷(𝒓) = 𝑪𝒓

𝒏𝒑𝒓𝒒𝒏−𝒓 n is the total number of trials, r is the success, p is the probability of success and q is the probability of failure. Example: Ellen takes a multiple choice quiz that has 5 questions. Each question has 4 answer choices. What is the probability that Ellen gets 3 correct by guessing? Since there are 5 questions – n=5 r = 3 correct answers Probability of guessing correctly - Probability of an incorrect guess – What is the probability that Ellen gets at least 3 by guessing? That means P(3) + P(4) + P(5) = Example: A water machine has a 98% probability of dispensing its product. If the machine dispenses 25 bottles and hour, what is the probability that there are 23 or fewer acceptable parts? Let’s calculate P(24) + P(25) and then subtract it from 1.

Homework for Binomial Distributions 1. Use the Binomial Theorem to expand:

a. (𝒙 + 𝟑)𝟒 b. (𝟐𝒂 − 𝒃)𝟑 2. The student council randomly chooses 6 students from the school population to represent the school in a homecoming assembly competition. The probability that an athlete is chosen is 30%. What is the probability that 4 athletes are chosen? 3. Tightwad Timmy bought 4 boxes of Colon-Blow Cereal. One out of every 5 boxes has a coupon for a free box of Colon-Blow. What is the probability that Tightwad gets at least 2 coupons? 4. A woman is expecting triplets. What is the probability that 2 are girls and 1 is a boy? 5. A tree has a 25% chance of flowering. In a random sample of 15 trees, what is the probability that at least 4 develop flowers? 6. An auto part has a 95% chance of being made within its tolerance level and a 5% chance of being defective. What is the probability that in a box of 8 parts, no more than 1 is defective?

PRE-CALCULUS PROBABILITY UNIT Two Way Tables

Two types of Data: ________________________ data can be described by a number, such as weight, length, age, etc.

________________________ data can be organized into categories such as sex, color, etc.

A _______ - _________ Table is a way of organizing data by two variables. Of the 300 students in the school, the number who went to Homecoming

Went to Homecoming Did not Go

Males 125 30

Females 135 10

Joint Relative Frequencies - Take each value and divide it by the total number (in this case, 300)

Went to Homecoming Did not Go Total

Males 0.42 0.10 .52

Females 0.45 0.03 .48

Total 0.85 0.13 1

Marginal Relative Frequencies – Found by adding each row across and each column down. Example: Tardy Ted is trying to decide on the best possible route to school. He has a choice of 3 routes. On each day, he randomly selects a route and keeps track of whether he is late or not. After a 40 day trial, his notes look like this: Late Not Late

Route A 4 10

Route B 3 7

Route C 4 12

Create a table of Joint and Marginal Relative Frequencies.

Late Not Late Total

Route A

Route B

Route C

Total 1

Find: P(Being late using Route A) P(Being late using Route B) P(Being late using Route C)

Homework for Two-Way Tables 1. Jenna has collected data on the number of students in the school who play a sport or plays a musical instrument. The following table shows her responses.

Plays a Sport

Yes No

Instrument Yes

48 28

Instrument No

76 48

a. Make a joint and marginal relative frequency table:

Plays a Sport

Yes No Total

Instrument Yes

Instrument No

Total

1

b. If you are given a student who plays a sport, what is the probability that the student also plays an

instrument? c. If you are given a student who plays an instrument, what is the probability that the student also

plays a sport? 2. The Customer Service Department at Auto Owners wants to evaluate how satisfied customers are with the handling of all of the insurance issues after a customer has experienced an accident. Auto Owners runs three departments that handle accident claims. The following is the results of a survey by customers:

Satisfied Dissatisfied

Department A 20 8

Department B 34 12

Department C 34 10

Make a joint and marginal relative frequency table based on the surveys.

Satisfied Dissatisfied Total

Department A

Department B

Department C

Total 1

a. Find the probability that a customer is satisfied after working with each department. P(Satisfied A) P(Satisfied B) P(Satisfied C) b. Which team had the highest rate of dissatisfaction is:

PRE-CALCULUS PROBABILITY UNIT Compound Events

A _______________________ event describes just a single outcome. A _______________________ event is made up of two or more simple events. ____________________________ ________________________ events are events that cannot __________ occur in the same trial of an experiment.

Mutually Exclusive Events 𝑷(𝑨 ∪ 𝑩) = 𝑷(𝑨) + 𝑷(𝑩)

Example: Pepsi is running a prize contest on the caps of the bottles. Under each cap is labeled “Free Pepsi”, “Free 2-Liter” or “Try Again”. There is a 1/10 probability that a “Free Pepsi” will appear and a 1/25 probability that a “Free 2-Liter” will appear. a. Explain why the events “Free Pepsi” and “Free 2-Liter” are mutually exclusive. b. What is the probability that a bottle cap is labeled “Free Pepsi” or “Free 2-Liter”.

Inclusive Events

_________________________ Events are events that have one or more ____________________ in common.

𝑷(𝑨 ∪ 𝑩) = 𝑷(𝑨) + 𝑷(𝑩) − 𝑷(𝑨 ∩ 𝑩)

(𝑷(𝑨 ∩ 𝑩) is the probability of both occurring) Example: A card is drawn from a deck of 52 cards.

a. Find the probability of drawing a king or a heart. b. Find the probability of drawing a red card or a face card (jack, queen, king).

PRE-CALCULUS PROBABILITY UNIT Two Way Tables

Mutually Exclusive Events

Event A Event B

Example: Of 3510 drivers surveyed, 1950 were male and 103 were color-blind. Only 6 of the color-blind drivers were female. What is the probability that a driver was male or was color-blind? Use a Venn Diagram: Total Drivers 𝑷(𝒎𝒂𝒍𝒆 ∪ 𝒄𝒐𝒍𝒐𝒓 − 𝒃𝒍𝒊𝒏𝒅) = 𝑷(𝒎𝒂𝒍𝒆) + 𝑷(𝒄𝒐𝒍𝒐𝒓 − 𝒃𝒍𝒊𝒏𝒅) − 𝑷(𝒎𝒂𝒍𝒆 ∩ 𝒄𝒐𝒍𝒐𝒓 − 𝒃𝒍𝒊𝒏𝒅)

Homework for Compound Events

1. Ten years after the graduation of 80 high school graduates, 50 had college degrees and 42 were married. Half of the students with a degree were married. Draw a Venn Diagram:

a. What is the probability that a student has a college degree or is married? b. What is the probability that a student has a college degree or is not married? c. What it the probability that a student does not have a college degree or is married? 2. The letters A-P are written on cards and placed in a box. Find the probability of each outcome. a. Choosing an E or choosing a G. b. Choosing an E or choosing a vowel. 3. Numbers 1-10 are written on cards and placed in a bag. Find each probability. a. Choosing a number greater than 5 or choosing an odd number. b. Choosing an 8 or choosing a number less than 5. c. Choosing at least one even number when selecting 2 cards from the bag. 4. On June 21st 2005, the Pistons were playing the San Antonio Spurs in game 7 of the NBA Finals. Nielsen claimed that 22% of all TV’s were tuned to the game. 15% were tuned into CSI.

a. What is the probability that someone who was watching TV that night was watching the Pistons or CSI?

b. Would this be considered a theoretical or experimental probability?

PRE-CALCULUS PROBABILITY REVIEW 1. How many different 7-digit telephone numbers can be made if the first digit cannot be 7, 8, 9 and 0? 2. From a group of 12 volunteers, a researcher must choose 5 to complete a study. How many groups of 5 people can be chosen? 3. A 5-digit code is given to all cashiers at Walmart to let them log onto the cash register. What is the probability that an employee receives a 5-digit code with all 5 numbers the same? 4. Find the probability that a point chosen inside the square is not inside the circle. 5. What is the probability of rolling 3 “doubles” in a row when rolling 2 dice? 6. The following shows the age and marital status of members of a local men’s horseshoe league.

18-34 35-50 51-65 66+

Married 6 20 22 4

Single 14 22 11 0

a. What is the probability the selected group is single, given that he is in the 35-50 group? b. What is the probability a married person 66 or older? c. What is the probability that a person in the group is single and in the 18-34 group? d. Complete the table below, giving the joint and marginal relative frequencies.

18-34 35-50 51-65 66+ Total

Married

Single

Total 1

10 cm

Binomial Distributions 7. A multiple-choice quiz has 5 questions. Each question has 3 possible answers. A lazy student has to guess the answer to each question. Find the probability that he guesses 3 out of the 5 questions correctly (thus passing the quiz). 8. Of 120 males and 180 females who took an eye exam, 170 passed. One third of the males did not pass. a. Make a Venn Diagram.

b. What is the probability that a person who took the exam passed or was male?