12-1: the counting principle learning targets: i can distinguish between independent and dependent...

12-1: The Counting Principle12-1: The Counting PrincipleLearning Targets:Learning Targets:

I can distinguish between independent and dependent events.

I can solve problems involving independent and dependent events.

The Counting PrincipleThe Counting PrincipleDefinitionsDefinitions

The Counting PrincipleThe Counting PrincipleDefinitionsDefinitions

trial: an experiment (like flipping a coin)

outcome: the result of a single trial

sample space: a list of all possible outcomes

event: one or more outcomes of a trial

The Fundamental The Fundamental Counting PrincipleCounting PrincipleThe Fundamental The Fundamental Counting PrincipleCounting Principle

If event M can occur in m ways and event N can occur in n ways, then event M followed by event N can occur in m•n ways.

-works with dependent events

-works with independent events

Dependent and IndependentDependent and IndependentEventsEvents

Dependent and IndependentDependent and IndependentEventsEvents

independent: the outcome of one event does not impact the outcome of another event

(rolling a die or tossing a coin)

dependent: the outcome of one event does impact the outcome of another event

(taking a sock out of a drawer and then taking another sock out of the same drawer without replacement of the first one)

Independent EventsIndependent EventsIndependent EventsIndependent Events

Make a tree diagram to see your sample space:

White Rye Cheese Bread

Butter Mustard Mayo Butter Mustard Mayo Butter Mustard Mayo

Nine possible combinations.

You could also do: Number of Breads ● Number of Spreads = 3 ● 3 = 9

A sandwich menu offers customers a choice of white, rye, or cheese bread with one spread chosen from butter, mustard, or mayonnaise. How many differentcombinations of bread and spread are there?

Independent EventsIndependent EventsIndependent EventsIndependent Events

Make a tree diagram to see your sample space:

Give a few minutes to complete.

A pizza place offers customers a choice of American,mozzarella, Swiss, feta, or provolone cheese with onetopping chosen from pepperoni, mushrooms orsausage. How many different combinations of cheeseand toppings are there?

Independent Events – Tree DiagramIndependent Events – Tree Diagram

American Mozzarella Swiss Feta ProvoloneAmerican Mozzarella Swiss Feta Provolone

Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau.Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau.

15 possible combinations.

You could also do: Number of Cheeses ● Number of Toppings = 5 ● 3 = 15

More Than Two Independent More Than Two Independent EventsEventsMore Than Two Independent More Than Two Independent EventsEvents

Communication How many codes are possible if ananswering machine requires a 2-digit code to retrieve messages?

Two Digit Code: ____ ____

How many digits can you choose from for each spot?

10 10

More Than Two Independent More Than Two Independent EventsEventsMore Than Two Independent More Than Two Independent EventsEvents

Communication How many codes are possible if ananswering machine requires a 2-digit code to retrieve messages?

Two Digit Code: _10_ ● _10_

There are 100 different codes to choose from.

PossibilitiesPossibilities

DigitsDigits 0-9 = 100-9 = 10

LettersLetters A-Z = 26A-Z = 26

CardsCards 52 total52 total

4 suits - 4 suits -

13 cards per suit13 cards per suit

Dependent EventsDependent EventsDependent EventsDependent Events

How many different schedules could a student havewho is planning to take 4 different classes? Assumeeach class is offered each period.

First Period Choices ● Second Period Choices ● Third Period Choices ● Fourth Period Choices

If a class is choosen for first hour, it can not been choosen again.

Dependent EventsDependent EventsDependent EventsDependent Events

How many different schedules could a student havewho is planning to take 4 different classes? Assumeeach class is offered each period.

4 ● 3 ● 2 ● 1 = 24

AssignmentAssignment

Work on the 21 problems that follow in the note packet.

Algebra 2A - Chapter 12Algebra 2A - Chapter 12Section 2Section 2

Permutations and Permutations and CombinationsCombinations

12-2: Permutations and Combinations12-2: Permutations and CombinationsLearning Targets:Learning Targets:

I can solve problems with permutations.

I can solve problems with combinations.

PermutationsPermutationsPermutationsPermutations

permutation: when a group of objects or people are arranged in a certain order

- order of objects very important

The number of permutations of n distinct objects taken r at a time is given by

P(n,r ) n!

(n r )!

Also written n Pr


P(n,r ) n!

(n r )!

Also written n Pr

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

Order Matters!!!


P(n,r ) n!

(n r )!

Also written n Pr

How many permutations of the letters MATH are possible?

Order Matters!!!


P(n,r ) n!

(n r )!

Also written n Pr

How many different four-letter code words can be

formed from the word EQUATIONS ?

Order Matters!!!

Also known as factorial:

Permutations Permutations with Repetitionwith RepetitionPermutations Permutations with Repetitionwith Repetition

You will notice some repetition here. The letter A appears thrice and the letter N appears twice.

How many different ways can the letters of the wordBANANA be arranged?

n!

p!q!

or

n!p!q! r!

The number of permutations of n objects of which p are alike and q are alike is:

CombinationsCombinationsCombinationsCombinations

combination: an arrangement or selection of objects in which order is not important

The number of combinations of n distinct objects taken r at a time is given by

C(n,r ) n!

(n r )! r!

Also written n Cr


C(n,r ) n!

(n r )! r!

Also written n Cr

Five cousins at a family reunion decide that three ofthem will go to pick up a pizza. How many ways canthey choose three people to go?

Order Does Not Matters!!!


C(n,r ) n!

(n r )! r!

Also written n Cr

There are 60 players on a football team. Seven of them will be chosen for a random drug test. How many ways can they be chosen?


Multiple EventsMultiple EventsMultiple EventsMultiple Events

C(n,r ) n!

(n r )! r!

Also written n Cr

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


There are 13 cards per suit.

Hearts Spades

Multiple EventsMultiple EventsMultiple EventsMultiple Events

C(n,r ) n!

(n r )! r!

Also written n Cr

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


There are 13 cards per suit.

Hearts Diamonds

How many of you have parentsthat play the Lottery?

Let’s calculate the number of differentcombinations there possibly are.

Mega Millions Total Combinations Since the total number of combinations for Mega Millions numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 56 is: COMBIN(56,5) = 3,819,816.

For each of these 3,819,816 combinations there are COMBIN(46,1) = 46 different ways to pick the sixth number (the “Mega” number). The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Mega Millions combinations is 3,819,816 x 46 = 175,711,536.

for Understandingfor Understanding

How many different ways can the letters of the wordALGEBRA be arranged?

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

Six friends at a party decide that three of them will goto pick up a movie. How many ways can they choose three people to go?


p. 641: 4-32

ReflectReflect

A class of 250 students wants to elect a committee of 4 to buy supplies for the homecoming float. How many different committees are possible?


ProbabilityProbability

12-3: Probability12-3: ProbabilityLearning Targets:Learning Targets:

I can find probability and odds of events.

I can create and use graphs of probability distributions.

ProbabilityProbabilityProbabilityProbability

success: desired outcome

failure: any outcome that is not a success

If an event can succeed in s ways and fail in f ways, then the probabilities of success P(S), and of failure, P(F), are as follows:

P(S) s

s f P(F)

f

s f


If an event can succeed in s ways and fail in f ways, then the probabilities of success P(S), and of failure, P(F), are as follows:

P(S) s

s f P(F)

f

s f

Probability is between 0 and 1, inclusive.

The closer to 1, the more likely the event is to occur. The closer to 0, the less likely the event is to occur.


When two coins are tossed, what is the probability that both are tails?

Use a sample space, tree diagram:

Toss #1: H T

Toss #2: H T H T

Probability with CombinationsProbability with CombinationsProbability with CombinationsProbability with Combinations

Monica has a collection of 32 CDs, of which 18 are R&B and 14 are rap. As she’s leaving for a trip, she grabs 6 CDs. What is the probability that she selects 3 R&B and 3 rap?

Combinations o

f R&B

Combinations o

f Rap

Probability with CombinationsProbability with CombinationsProbability with CombinationsProbability with Combinations

Roman has a collection of 26 books–16 are fiction and10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability thathe chooses 4 fiction and 4 nonfiction?

OddsOddsOddsOdds

The odds that an event will occur can be expressed as the ratio of the number of ways it can succeed to the number of ways it can fail. If an event can succeed in s ways and fail in f ways, then the odds of success and of failure are as follows:

Odds of success = s : f

Odds of failure = f : s

Notice: s + f = Total Possibilities

OddsOddsOddsOdds

According to the CDC, the chances of a male born in 1990 living to age 65 are about 3 in 4. For females the chances are about 17 in 20.

What are the odds of a male living to be at least 65?

What are the odds of a female living to be at least 65?

3:1Success

Failure:

4 -3

17:3

Probability DistributionsProbability DistributionsWhich outcomes are least likely? most likely?Which outcomes are least likely? most likely? Probability DistributionsProbability DistributionsWhich outcomes are least likely? most likely?Which outcomes are least likely? most likely?

Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution ofthe sum of the numbers rolled.

Probability

12111098765432S = Sum


When three coins are tossed, what is the probabilitythat all three are heads?

Life Expectancy The chances of a male born in 1980to live to be at least 65 years of age are about 7 in 10.For females, the chances are about 21 in 25. Calculate the odds for each sex living at least 65 years.


p. 647: 4-18

p. 648: 19-53

ReflectReflect

If 7 out of 8 students prefer the subject of math to literature, what are the odds that students prefer math? that students prefer literature?


MultiplyingMultiplying

ProbabilitiesProbabilities

12-4: Multiplying Probabilities12-4: Multiplying ProbabilitiesLearning Targets:Learning Targets:

I can find the probability of two independent events.

I can find the probability of two dependent events.

Probability RulesProbability RulesProbability RulesProbability Rules

Probability of two independent events:

P(A and B) = P(A) • P(B)

Probability of two dependent events:

P(A and B) = P(A) • P(B following A)

extends to

P(A, B, C) = P(A) • P(B following A) • P(C following A and B)

Independent EventsIndependent Events

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

regular soft drink?

This is a problem With Replacement!!

Independent EventsIndependent Events

Gernardo 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 of the coins he selects are dimes?

This is a problem With Replacement!!

Independent Events ExtendedIndependent Events Extended

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?

P(6) P(6) P(not 6) =

Three Independent EventsThree Independent EventsThree Independent EventsThree Independent Events

When three dice are rolled, what is the probabilitythat two dice show a 5 and the third die shows aneven number?

P(5) P(5) P(even) =

Two DEPENDENT EventsTwo DEPENDENT EventsTwo DEPENDENT EventsTwo DEPENDENT Events

In the previous Julio and the soft drink example, what is the probability that both people select a regular soft drink if Julio does NOT put his drink back into the cooler?

This is a problem Without Replacement!!

Two Dependent EventsTwo Dependent EventsTwo Dependent EventsTwo Dependent Events

The host of a game show is drawing chips from a bag to determine prizes. Of the 10 chips in the bag, 6 show TV, 3 show VACATION, and 1 shows CAR. If the host draws the

chips at random without replacement, find the probabilities:

a. a vacation, then a car

b. two TVs

Three Dependent EventsThree Dependent EventsThree Dependent EventsThree Dependent Events

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

for Understanding #1for Understanding #1

When three dice are rolled, what is the probabilitythat one die is a multiple of 3, one die shows aneven number, and one die shows a 5?

P(x3) P(even) P(5) =


The host of a game show draws chips from a bag to determine the prizes for which contestants will play. Of the 20 chips in the bag, 11 show computer, 8 show trip, and 1 shows truck. If the host draws the chips at random and does not replace them, find each probability.

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.


p. 654: 4-12 even

p. 655: 14-34 even

p. 656: 40, 42

ReflectReflect

When four dice are rolled, what is the probabilitythat two dice show a 3 and the third die shows aneven number?


AddingAdding

ProbabilitiesProbabilities

12-5: Adding Probabilities12-5: Adding ProbabilitiesLearning Targets:Learning Targets:

I can find the probability of mutually exclusive events.

I can find the probability of inclusive events.

What are mutually exclusive What are mutually exclusive events? events? What are mutually exclusive What are mutually exclusive events? events?

simple event: only one event (like rolling a 1)

compound event: two or more simple events (like rolling an odd or a 6)

mutually exclusive events: events that cannot occur at the same time (like when you consider the prob of drawing a 2 or an ace---you can’t draw a 2 and an ace at the same time, drawing a 2 and an ace are said to be mutually exclusive events)

Probability of Mutually Probability of Mutually Exclusive Events Exclusive Events Probability of Mutually Probability of Mutually Exclusive Events 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)

This can be extended to any number of mutually exclusive events.

Two Mutually Two Mutually Exclusive EventsExclusive EventsTwo Mutually Two Mutually Exclusive EventsExclusive Events

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

P(base or soc) = P(base) + P(soc)

P(base or soc) =

Three Mutually Three Mutually Exclusive EventsExclusive EventsThree Mutually Three Mutually Exclusive EventsExclusive Events

There are 7 girls and 6 boys on the homecoming committee. A subcommittee of four 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?

P(at least 2 g) = P(2 girls) + P(3 girls) + P(4 girls)

What are Inclusive Events?What are Inclusive Events?What are Inclusive Events?What are Inclusive Events?

Inclusive events are ones whose outcomes may be the same. They are NOT mutually exclusive

Example:

drawing a queen or a diamond

Q o

f D

Probability of Inclusive Probability of Inclusive EventsEventsProbability of Inclusive Probability of Inclusive EventsEvents

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

P(A or B) = P(A) + P(B) - P(A and B)

Back to our Back to our Queen of DiamondsQueen of DiamondsBack to our Back to our Queen of DiamondsQueen of Diamonds

P(queen or diamond) =

= P(Queen) + P(Diamond) - P(Queen of Diamonds)

Inclusive EventsInclusive EventsInclusive EventsInclusive Events

The enrollment at South High School is 1400. Suppose 550 students take French, 700 take algebra, and 400 take both French and algebra. What is the probability that a student selected at random takes French or algebra?

= P(French) + P(Algebra) - P(Both)


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

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


p. 660-661: 4-16 all, 17-31 odd

ReflectReflect

There are 200 students taking Calculus, 500 taking Spanish, and 100 taking both. There are 1000 students in the school. What is the probability that a student selected at random is taking Calculus or Spanish?

Algebra 2A - Chapter 12Algebra 2A - Chapter 12Review Sections 1-5Review Sections 1-5

Probability.Probability.

12-1: the counting principle learning targets: i can distinguish between independent and dependent...

Documents

independent eventsmake

event n

different codes

independent eventsdependent

different schedules

different classes

digit code

period choicesif