12.1 The Counting Principle

Vocabulary

Outcome: the result of a single trial

Sample Space: set of all possible outcomes

Event: one or more outcomes of a trial

Independent Events: choice of one thing DOES NOT affect the choice of another

Dependent Events: choice of one thing DOES affect the choice of another

The Fundamental Counting Principle

If one event can occur m ways and another event can occur in n ways, then one event followed by the other event can occur in mn ways

Basically multiply the number of ways for each event to get the total number of ways the events can occur together

Examples

A sandwich cart offers the choice of hamburger, chicken or fish on plain or sesame bun. How many combinations of meat and bun are possible? Note: meat choice does not affect bun

choice so these events are independent

Kim won a contest on the radio. The prize was a restaurant gift certificate and tickets to a sporting event. She can select one of three restaurants and tickets for football, baseball, basketball, or hockey game. How many different ways can she select a restaurant followed by a sporting event?

A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayonaise. How many bread and spread combinations are possible?

The Murray’s are choosing a trip to the beach or the mountains. They can travel by car, train, or plane. How many ways can the family select a trip followed by means of transportation?

How many answering machine codes are possible if the code is two digits?

How many license plates can be made if the first three places must be letters and the last three must be numbers?

How many area codes are possible if each area code is 3-digits?

How many ATM pin numbers are there if each pin number is 4 characters long and each character could be a number or a letter?

Charlita wants to take 6 different classes next year. Assuming that each class is offered each period, how many different schedules could she have? Note if she takes Algebra II first period

she won’t take it another period… so this is a dependent event.

Each player in a board game uses one of six different pieces. If four players play the game, how many different ways could the players choose their game pieces?

An ice cream shop offers a choice of two types of cones and 15 flavors of ice cream, and the choice of peanuts, chocolate sprinkles, or crushed oreos for toppings. How many different 1-scoop, 1-topping cones can a customer order?

12.2 Permutations & Combinations

Factorial

if n is a positive integer, then n! = n x (n – 1) x (n – 2) x ….

Any number with a ! behind it is a factorial

Permutations

When a group of objects or people are arranged in a certain order.

(order matters)

* Also written as nPr

Examples

Eight people enter the Best Pie contest. How many ways can blue, red, and green ribbons be awarded?

Ten people are competing in a swim race where 4 ribbons will be given. How many ways can blue, red, green, and yellow ribbons be awarded?

Permutations with Repetitions

n = total number

p & q = the number of times each thing repeats

Examples

How many different ways can the letters of the word BANANA be arranged?

How many different ways can the letters of the word ALGEBRA be arranged?

Combinations

An arrangement or selection of objects in which order is not important

* Also written as nCr

Examples

Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose three people to go?

Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades?

Thirteen cards are drawn from a standard deck of cards. How many hands consist of six hearts and seven diamonds?

A coach must choose five starters from a team of 12 players.  How many different ways can the coach choose the starters?

If 20 people work in a an office and 4 are selected to go to a conference how many different selections are possible?

If the Junior class is voting on class officers and 8 people have volunteered for the positions of President, Vice President, and Historian how many ways can the students select their class officers?

In gym class Blake is picking a team for tennis, he needs to pick 3 people from his class of 20. How many different teams could he form?

12.3 Probability

Vocabulary

Probability: a ratio that measures the chances of an event occurring.

Success: a desired outcome

Failure: any other outcome

Odds: a ratio of success to failures (odds of success) a ratio of failures to success (odds of failure)

Vocabulary Continued

Random: when all outcomes have an equally likely chance of occuring

Random Variable: a variable whose value is the numerical outcome of a random event

KEY CONCEPTS

Probability of Success: If an event can succeed in s ways (will occur)

Probability of Failure: If an event can fail in f ways (will not occur)

fs

sSP

)(

fs

fFP

)(

Examples

What’s the probability of flipping a coin and having it land on heads?

Examples

Find the odds of an event occurring, given the probability of an event.

81

125

13

6

81

125

13

6

Examples

Find the probability of an event occurring, given the odds of the event

9:8

5:4

1:12

If there are 18 marbles in a bag and 3 are red and 4 are green, 3 are white, and 8 are blue what’s the probability of choosing:

A red marble

A green marble

Not picking a white marble

Not picking a blue marble

Probability with Combinations and Permutations

Follow these steps:1. Write the combination or permutation

for the first group

2. Multiply by the combination or permutation for the second group

3. Divide the product by the total combinations or permutations possible

4. Write a fraction for the probability

Monica has a collection of 32 CD’s- 18 R&B and 14 rap. As she is leaving for a trip, she randomly chooses 6 CD’s to take with her. What is the probability that she selects 3 R&B and 3 rap?

A board game is played with tiles and letters on one side. There are 56 tiles with consonants and 42 tiles with vowels. Each player must choose seven of the tiles at the beginning of the game. What is the probability that a player selects four consonants and three vowels?

Ramon has five books on the floor, one for each of his classes: Algebra 2, Chemistry, English, Spanish, and History. Ramon is going to put the books on a shelf. If he picks the books up at random and places them ina row on the same shelf, what is the probability that his English, Spanish, and Algebra 2 books will be the leftmost books on the shelf, but not necssarily in that order?

For next semester, Alisa has signed up for English, Precalculus, Spanish, Geography, and chemistry classes. If class schedules are assigned randomly and each class is equally likely to be at any time of day what is the probability that Alisa’s first two classes in the morning will be Precalculus and Chemistry, in either order?

12-4 Multiplying Probabilities

Probability of Two Independent Events

If two events A and B are independent then the probability of both events occurring is

P(A and B) = P(A) ∙ P(B)

*The denominator should not change

Examples

At a picnic, Julio reaches into an ice-filled cooler containing 8 regular soft drinks and 5 diet soft drinks. He removes a can, then decides he is not really thirsty, and puts in back. What is the probability that Julio and the next person to reach into the cooler both randomly select a regular soft drink?

Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both coins he selects are dimes?

In a board game, three dice are rolled to determine the number of moves for the players. What is the probability that the first die shows a 6, the second die shows a 6, and the third die does not?

When three dice are rolled, what is the probability that the first two show a 5 and the third shows an even number?

In a state lottery game, each of three cages contains 10 balls. The balls are each labeled with one of the digits 0-9. What is the probability that the first two balls drawn will be even and that the third will be prime?

Probability of Two Dependent Events

If two events A and B are dependent, then the probability of both events occurring is

P(A and B) = P(A) ∙ P(B following A)

* The denominator can/should change

Back to Example 1 with Julio

What is the probability that both people select a regular soft drink if Julio does not put his back in the cooler?

The host of a game show is drawing chips from a bag to determine the prizes for which contestants will play. Of the 10 chips in the bag, 6 show television, 3 show vacation, and 1 shows car. If the host draws the chips at random and does not replace them, find the probability that he draws a

vacation, then a car.

Use the information above. What is the probability that the host draws two televisions?

The host of a game show is drawing chips from a bag to determine the prizes for which contestants will play. Of the 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. If chips are drawn at random and without replacement, find the probability of drawing a computer, then a truck.

Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a heart, another heart, and a spade in that order.

Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a diamond, a club, and another diamond in that order.

Find the probability of drawing three cards of the same suit.

12-5 Adding Probabilities

Vocabulary

Simple Event: cannot be broken down into smaller events Rolling a 1 on a 6 sided die

Compound Event: can be broken down into smaller events Rolling an odd number on a 6 sided die

Mutually Exclusive Events: two events that cannot occur at the same time Drawing a 2 or an ace from a deck of cards

A card cannot be both a 2 and an ace

Probability of Mutually Exclusive Events

If two events A and B, are mutually exclusive, then the probability that A or B occurs is the sum of their probabilities. P(A or B) = P(A) + P(B)

Example: Find the probability of rolling a 2 or an ace from a deck of cards.P(2 or ace) = P(2) + P(ace)

4/52 + 4/52 8/52 2/13

Examples

Keisha has a stack of 8 baseball cards, 5 basketball cards, and 6 soccer cards. If she selects a card at random from the stack, what is the probability that it is a baseball or a soccer card?

One teacher must be chosen to supervise a senior class fundraiser. There are 12 math teachers, 9 language arts teachers, 8 social studies teachers, and 10 science teachers. If the teacher is chosen at random, what is the probability that the teacher is either a language arts teacher or a social studies teacher?

There are 7 girls and 6 boys on the junior class homecoming committee. A subcommittee of 4 people is being chosen at random to decide the theme for the class float. What is the probability that the subcommittee will have at least 2 girls?

More Vocabulary

Inclusive Events: when two events are not mutually exclusive Example Picking a King or a Spade

It is possible to have one card that is both King and Spade

Let’s think about this…

Probability of Inclusive Events

If two events A and B are inclusive, then the probability that A or B occurs in the sum of their probabilities decreased by the probability of both occurring

P(A or B) = P(A) + P(B) – P(A and B)

Suppose that of 1400 students , 550 take Spanish, 700 take biology, and 400 take both Spanish and biology. What is the probability that a student selected at random takes Spanish or biology?

Sixty plastic discs, each with one of the numbers from 1 to 60, are in a bag. LaTanya will win a game if she can pull out any disc with a number divisible by 2 or 3. What is the probability that LaTanya will win?

The Cougar basketball team can send 5 players to a basketball clinic. Six guards and 5 forwards would like to attend the clinic. If the players are selected at random, what is the probability that at least 3 of the players selected to attend the clinic will be forwards?

There are 2400 subscribers to an Internet service provider. Of these, 1200 own desktop computers, 500 own laptop computers, and 100 own both a desktop and a laptop. What is the probability that a subscriber selected at random owns either a desktop or a laptop?

In the Math Club, 7 of the 20 girls are seniors, and 4 of the 14 boys are seniors. What is the probability of randomly selecting a boy or a senior to represent the Math Club at a statewide math contest?

Sylvia has a stack of playing cards consisting of 10 hearts, 8 spades, and 7 clubs. If she selects a card at random from the stack, what is the probability that it is a heart or a club?

12.6 Statistical Measures

Measures of Central Tendency

Vocabulary

Univariate Data: data with one variable

Measure of central tendency: one number that describes a set of data

USE WHEN…

mean The data are spread out and you want an average (add all and divide by total # of items)

Median The data contains outliers (put all in order from least to greatest and find the exact middle number)

Mode The data are tightly clustered around one or two values (the #s that appear the most often)

Measures of Variation (dispersion)

Measure how spread out or scattered a set of data is Simplest measure of variation to

calculate is the range

Variance and Standard deviation: measures of variation that indicate how much the data values differ from the mean

To find variance (σ2)

1. Find the mean2.Find difference between each value in the

set of data and the mean3. Square each difference4. Find the mean of the squares

Standard deviation (σ ) is the square root of the variance

x

Key Concept

n

xxxxxxxx n

22

3

2

2

2

1

The equation below is the formula for standard deviation.

Examples

Data: 1, 2, 2, 3, 3, 3, 4, 4, 5

Mean Stem and Leaf Box and whisker

Median

Mode

{234, 345, 123, 368, 279, 876, 456, 235, 333, 444}

Mean Stem and Leaf Box and whisker

Median

Mode

A sweepstakes offers a first prize of $10,000, two second prizes of $100, and one hundred third prizes of $10. Which measure of central tendency best represents the available prizes?

Which measure of central tendency would the organizers of the sweepstakes be most likely to use in their advertizing?

The leading number of home runs in Major League Baseball for the 1994-2004 seasons were 43, 50, 52, 56, 70, 65, 50, 73, 57, 47, and 48. Find the variance and standard deviation of the data to the nearest tenth.

The following list shows the rebounding totals for the members of the 2005 Charlotte Sting: 162, 145, 179, 37, 44, 53, 70, 65, 47, 35, 71, 5, 5. Find the variance

and standard deviation.