Transcript 12.1 The Counting Principle

12.1
The Counting Principle
Vocabulary
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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
Outcome: the result of a single trial
Sample Space: set of all possible
outcomes
Event: one or more outcomes of a trial
The Fundamental Counting Principle
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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
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Basically multiply the number of ways
for each event to get the total number
of ways the events can occur together
Examples
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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?
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Note: meat choice does not affect bun
choice so these events are independent
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An ice cream shop offers two types
of cones or a bowl, 17 flavors of ice
cream, and 40 toppings. How many
combinations of one-scoop, onetopping sundaes can you order?
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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?
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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?
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How many answering machine
codes are possible if the code is two
digits?
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How many license plates can be
made if the first three places must
be letters and the last three must
be numbers?
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How many area codes are possible
if each area code is 3-digits?
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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?
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Charlita wants to take 6 different
classes next year. Assuming that
each class is offered each period,
how many different schedules could
she have?
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Note if she takes Algebra II first period
she won’t take it another period… so
this is a dependent event.
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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?
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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
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if n is a positive integer, then
n! = n x (n – 1) x (n – 2) x ….
Any number with a
factorial
! behind it is a
Permutations
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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?
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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?
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How many different ways can the
letters of the word ALGEBRA be
arranged?
Combinations
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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?
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Thirteen cards are drawn from a
standard deck of cards. How many
hands consist of six hearts and
seven diamonds?
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A coach must choose five starters
from a team of 12 players. How
many different ways can the coach
choose the starters?
Mixed Examples: Identify as permutation,
permutation w/ rep. or combination
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If 20 people work in a an office and 4
are selected to go to a conference
how many different selections are
possible?
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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?
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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?
12.3
Probability
Vocabulary
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Probability: a ratio that measures the
chances of an event occurring.
Odds:
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a ratio of success to failures (odds of success)
a ratio of failures to success (odds of failure)
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Failure: any other outcome
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Success: a desired outcome
Vocabulary Continued
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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
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Probability of Success: If an event
can succeed in s ways (will occur)
s
P( S ) 
s f
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Probability of Failure: If an event can
fail in f ways
(will not occur)
f
P( F ) 
s f
Examples
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What’s the probability of flipping a
coin and having it land on heads?
Examples
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6
Find the odds of an event occurring,
given
13
the probability of an event.
6
13
5
12
1
8
5
12
1
8
Examples
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Find the probability of an event
occurring, given the odds of the event
12 : 1
4:5
8:9
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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:
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A red marble
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A green marble
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Not picking a white marble
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Not picking a blue marble
Probability with Combinations and
Permutations
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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
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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
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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?
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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?
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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?
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When three dice are rolled, what is the
probability that the first two show a 5 and
the third shows an even number?
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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
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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
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What is the probability that both people
select a regular soft drink if Julio does not
put his back in the cooler?
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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.
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Use the information above. What is
the probability that the host draws
two televisions?
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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.
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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.
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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.
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Find the probability of drawing three cards
of the same suit.
12-5
Adding Probabilities
Vocabulary
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Simple Event: cannot be broken down
into smaller events
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Compound Event: can be broken down
into smaller events
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Rolling a 1 on a 6 sided die
Rolling an odd number on a 6 sided die
Mutually Exclusive Events: two events
that cannot occur at the same time
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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
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If two events A and B, are mutually
exclusive, then the probability that A or B
occurs is the sum of their probabilities.
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P(A or B) = P(A) + P(B)
Examples
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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?
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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?
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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
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Inclusive Events: when two events
are not mutually exclusive
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Example Picking a King or a Spade
It is possible to have one card that is
both King and Spade
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Let’s think about this…
Probability of Inclusive Events
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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
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P(A or B) = P(A) + P(B) – P(A and B)
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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?
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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?
Mixed Examples: Identify as
mutually exclusive or inclusive
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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?
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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?
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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?