Who am I?

Johann Carl Friedrich Gauss1777 – 1855

Friday’s Benchmark• Scores were typical

• 1st Block - ~30% scored 60 or better• 2nd Block - ~25% scored 60 or better

• Today we start probability

Warm Up Part 1

Warm Up Part 2

And now . . .

Probability!!

First . . .

Let’s Play A Game

Definition - Probability The likelihood that an event occurs

Represented as a fraction, decimal, or percent

(As a decimal) Cannot be less than zero or greater than 1

Experimental v Theoretical Experimental probability is what is observed in a series of trials (i.e., a sample)

Theoretical probability is what is calculated mathematically

Applications of Probability Probabilities are used to help predict the future or better understand a large (sometimes immeasurable) population

Ex 1 What is the probability that a blue marble is randomly selected from a bag containing seven red, four yellow, and three blue marbles? (record your answer as a fraction)

Ex 2 What is the probability that a blue marble is not randomly selected from a bag containing seven red, four yellow, and three blue marbles? (record your answer as a percent rounded to the nearest tenth of a percent)

The Compliment of an Event The compliment of an event is the set of all other events that aren’t the one in question

e.g., • The compliment of a coin toss resulting in tails is…• The compliment of it raining tomorrow is…• The compliment of the Carolina Panthers being the

second team in NFL history to have a perfect season is …

Ex 3 During a five week period the duration of the morning announcements was recorded everyday. In that time there were 7 mornings that had announcements which exceeded 10 minutes. a) Use this data to calculate a the probability that the announcements will not be longer than 10 minutes tomorrow? b) Is the probability calculated in part a a theoretical or experimental probability? c) Approximately how many days next week would you expect there to be morning announcements lasting more than 10 minutes?

Ex 4 Two six-sided number cubes (i.e., dice) are rolled at the same time. What is the probability that the sum of the two resulting faces are 10 or more?

Who am I?

Gary Wayne Coleman1968 – 2010

Warm Up Part 1

And now . . .

Probability!!

Review From YesterdayWhat is a probability?

How do we represent them?

What restrictions/conditions are there?

How do we calculate it? (Raw definition)

Review From YesterdayWhat’s an event?

What’s the complement of an event?

What must the sum of the probability of an event and the probability of the event’s complement be?

Review Ex 1What’s the probability of rolling a number less than 3 when using a fair six-sided number cube? (i.e., P(less than 3)) Record answer as a fraction in reduced form.

Review Ex 2What’s the probability that a student, randomly selected, from this class room is male? Record answer as a percent rounded to the tenth.

Review Ex 3What’s the probability of drawing a King from a standard deck of cards? Record answer as a decimal rounded to the thousandths.

Review Ex 4What’s the probability of drawing a seven or a heart from a standard deck of playing cards? Record answer as a fraction in simplest terms.

Review – You TryRecord answers as percents rounded to the tenth.

1) What’s the probability of rolling an odd number using a fair six-sided number cube?

2) What’s the probability of drawing black card or an Ace from a standard deck of playing cards?

Back to where we left off . . .

Probability!!

Ex 5 Three marbles are randomly selected (at one time) from a bag containing a total of 20 marbles. The bag has nine blue marbles and 11 marbles that aren’t blue. What is the probability that all of them are blue?

Ex 6 Three marbles are randomly selected (one at a time, and then replaced after drawing) from a bag containing a total of 20 marbles. The bag has nine blue marbles and 11 marbles that aren’t blue. What is the probability that all of them are blue?

Rep v No Rep Rep is the short hand I use for “Repetition” or “Replacement”

This describes the concept of a specific (i.e., unique) event can occur more than one time

Ex 7 Three marbles are randomly selected (one at a time, and then replaced after drawing) from a bag containing a total of 20 marbles. The bag has nine blue marbles and 11 marbles that aren’t blue. What is the probability that exactly two of them are blue?

You Try During the winter of 2015-2016 (which lasted for 92 days) it snowed a total of 48 days in the city of Oswego, NY. Use this data to answer the following questions. a) What is the probability that it snows in Oswego on Christmas day? (Record answer as a decimal rounded to the thousandths) b) Is the probability you determined in part a a theoretical or experimental probability? c) What is the probability it snows on Christmas Eve and Christmas day in Oswego? (Record answer as a decimal rounded to the thousandths)

Who am I?

Todd Anthony BridgesBorn 1965

Warm Up

Warm Up540 total

308 people prefer chocolate

263 people were female

152 people were men who prefer vanilla

Warm Up540 total

308 people prefer chocolate 232 people prefer vanilla

263 people were female 277 people were male

152 people were men who prefer vanilla

Warm Up540 total

308 people prefer chocolate 232 people prefer vanilla

263 people were female 277 people were male

152 people were men who prefer vanilla

Warm Up

Warm Up

Warm Up (of probability stuff)A random sample of 112 students at NC state were surveyed as to whether or not they were natives of NC. Of these students 81 were from NC and the rest were from somewhere outside of the state.a) What’s the probability that a randomly selected student

from NC state is a native of a state that isn’t NC?b) How many NC residents would you expect there to be in a

physics class at NCSU of 32 people?

Warm Up (of probability stuff)A random sample of 112 students at NC state were surveyed as to whether or not they were natives of NC. Of these students 81 were from NC and the rest were from somewhere outside of the state.a) What’s the probability that a randomly selected student

from NC state is a native of a state that isn’t NC? 0.277b) How many NC residents would you expect there to be in a

physics class at NCSU of 32 people? 23 or 24

Counting Methods

A Sub-Topic of Probability

Outline• Counting is important and needs to be accurate and efficient

• We will discuss the counting of arrangements (also called orderings or placements)

Arrangements• Permutation –

• Combination –

Arrangements• Permutation – an arrangement in which the order of items (or elements) is important (e.g., spelling of a name)

• Combination –

Arrangements• Permutation – an arrangement in which the order of items (or elements) is important (e.g., spelling of a name)

• Combination – an arrangement in which the order of items (or elements) is unimportant (e.g., toppings on a pizza)

Arrangements• With repetition –

• Without repetition –

Arrangements• With repetition – items (or elements) can be repeated in the arrangement

• Without repetition –

Arrangements• With repetition – items (or elements) can be repeated in the arrangement

• Without repetition – items (or elements) can not be repeated in the arrangement

Formulas•Permutation with repetition:• There are n items to choose FROM and r items

must CHOSEN

•Permutation without repetition:• There are n items to choose FROM and r items

must CHOSEN nPr (where P is a function)

Formulas•Combination with repetition:• There are n items to choose FROM and r items

must CHOSEN (n+r-1)Cr

•Combination without repetition:• There are n items to choose FROM and r items

must CHOSEN nCr (where C is a function)

Ex 1 How many different nine man batting orders are possible on a baseball team of nine people?

Ex 1 How many different nine man batting orders are possible on a baseball team of nine people?

9P9 = 362,880

Ex 2 Dominoes has a total of 17 different toppings. How many unique types of three topping pizzas can be ordered? (Assume toppings can be ordered more than once)

Ex 2 Dominoes has a total of 17 different toppings. How many unique types of three topping pizzas can be ordered? (Assume toppings can be ordered more than once)

19C3 = 969

Ex 3 Giacomo owns 16 suits and needs to select three of them for a business conference he must attend. How many different sets of three suits can he bring with him?

Ex 3 Giacomo owns 16 suits and needs to select three of them for a business conference he must attend. How many different sets of three suits can he bring with him?

16C3 = 560

Ex 4 How many different five card poker hands are possible when dealing from a standard deck of 52 cards?

Ex 4 How many different five card poker hands are possible when dealing from a standard deck of 52 cards?

52C5 = 2,598,960

Ex 5 Jennifer has seven t-shirts, five pairs of gym shorts, three pairs of sneakers, and 39 sweat bands. How many different work out outfits does she have?

Ex 5 Jennifer has seven t-shirts, five pairs of gym shorts, three pairs of sneakers, and 39 sweat bands. How many different work out outfits does she have?

7 * 5 * 3 * 39 = 4095

Ex 6 A senate sub-committee is commonly comprised of seven senators. How many unique sub-committees are possible with the 102 members of the US senate?

Ex 6 A senate sub-committee is commonly comprised of seven senators. How many unique sub-committees are possible with the 102 members of the US senate?

102C7 = 18,466,953,120

Ex 7 A quiz is made up of ten “True or False” questions. How many different ways can the quiz be randomly answered?

Ex 7 A quiz is made up of ten “True or False” questions. How many different ways can the quiz be randomly answered?

210 = 1024

Ex 8 An 8 question multiple choice test has options of A, B, C, or D for all items on the test. If all questions are randomly answered what is the probability that all questions are answered correctly? (Record your answer as a percent rounded to the nearest hundredth of a percent)

Ex 8 An 8 question multiple choice test has options of A, B, C, or D for all items on the test. If all questions are randomly answered what is the probability that all questions are answered correctly? (Record your answer as a percent rounded to the nearest hundredth of a percent)

48 = 65536

You Try A MathXL HW assignment is made up of five “True or False” questions. If all questions are randomly answered what is the probability the score on the HW is a 0? (Assume the HW is attempted and no questions are answered correctly)

You Try A MathXL HW assignment is made up of five “True or False” questions. If all questions are randomly answered what is the probability the score on the HW is a 0? (Assume the HW is attempted and no questions are answered correctly)

You Try Frank’s wife Angela tells him to order Dominoes for dinner. She wants a large pizza with three different toppings on it. Angela is very particular (and difficult) she only likes two different types of three-topping pizzas from Dominoes and she’s never told Frank what they are. If Dominoes offers 9 toppings and we know Angela doesn’t like to repeat toppings, what is the probability that Frank orders a pizza that pleases his wife’s palate?

You Try Frank’s wife Angela tells him to order Dominoes for dinner. She wants a large pizza with three different toppings on it. Angela is very particular (and difficult) she only likes two different types of three-topping pizzas from Dominoes and she’s never told Frank what they are. If Dominoes offers 9 toppings and we know Angela doesn’t like to repeat toppings, what is the probability that Frank orders a pizza that pleases his wife’s palate?

Fast Fact Both the permute and “combination” functions (i.e., nPr and nCr) involve what’s known as the factorial operator

Factorial

e.g.,

Go to:www.mathxlforschool.com

Assignment: Probability 1Due: Tomorrow 11:59pm