Name:
Period:
Honors Geometry
UNIT 12 NOTES: Probability
1
Honors Geometry
Unit 12 Plan: This plan is subject to change at the teacher’s discretion.
Section
Topic
Assignment
12.1a
Basic Concepts in Probability
12.1a WS
12.1b
The Complement Rule
12.1b WS
12.2
Conditional Probability
12.2 WS
12.3
Multiplication Rule
12.3 WS
Review
12.1 – 12.2 Review
Finish Review
Quiz
12.1 – 12.2 Quiz
None
12.4
The Addition Rule
12.4 WS
12.5a
Permutations
12.5a WS
12.5b
12.5b WS
12.6
Combinations & Applications of
Counting Principles
Geometric Probability
Review
Review Unit 12
Finish Review
Test
Test Unit 12
None
Due Date
12.6 WS
UNIT 12 TEST DAY:
2
12.1a: Basic Concepts of Probability Day 1
Objectives:

Identify the sample space of a probability experiment.

Use a tree diagram and the Fundamental Counting Principle to find probabilities.

Identify simple events
Notes:
Probability experiment
•
An action, or trial, through which specific results (counts, measurements, or responses) are
obtained.
Outcome
•
The result of a single trial in a probability experiment.
Sample Space
•
The set of all possible outcomes of a probability experiment.
Event
•
Consists of one or more outcomes and is a subset of the sample space.
Practice:
1. A probability experiment consists of rolling two dice and adding them to get a total.
 Probability experiment:
 Outcome:
 Sample space:
 Event:
 Uniform or Not Uniform:
2. A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the
sample space using a tree diagram. Is this uniform or not?
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3. Mr. Puetz is getting dressed for school – he can choose from a white shirt, blue shirt, or red shirt. He
can choose from tan pants or navy pants. He can choose from black shoes or brown shoes. Describe
the sample space of Mr. Puetz’s wardrobe (Note: some outfits may be better than others)
4. What is the probability of exactly four girls in families of four children?
a) How many ways are there to have four children?
b) How many of these are all girls?
c) What is the P(4 girls)?
5. You are at a carnival. One of the carnival games asks you to pick a door and then pick a curtain
behind the door. There are 3 doors and 4 curtains behind each door. How many choices are possible
for the player?
4
6. The 4 aces are removed from a deck of cards. A coin is tossed and one of the aces is chosen. What
is the probability of getting heads on the coin and the ace of hearts? Draw a tree diagram to illustrate
the sample space.
Notes: Simple Event
Simple Event:
– ex.
An event that consists of more than one outcome is not a simple event.
– ex.
Practice: Determine whether the event is simple or not.
1. You reach into a deck of cards and randomly pull out one. Event A is pulling out the Ace of Spades.
2. You reach into a deck of cards and randomly pull out one. Event B is pulling out a face card.
3. For quality control, you randomly select a machine from a batch that has been manufactured that
day. Event C is selecting a specific defective machine part.
4. You roll two dice. Event D is rolling a total of 7.
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Notes: Fundamental Counting Principle
•
Use when an event can occur in so many ways that it is not practical to write out all of the
outcomes.
•
If one event can occur in m ways and a second event can occur in n ways, the number of ways
the two events can occur in sequence is m∙n.
•
Can be extended for any number of events occurring in sequence.
5. A probability experiment consists of buying a car. This car comes in the colors of Red, Blue, and
Green. Convertible or hard top, Chrome, Steal or Aluminum wheels. How many ways can I buy this
car.
6. You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed.
Manufacturer: Ford, GM, Honda
Car size: compact, midsize
Color: white (W), red (R), black (B), green (G)
How many different ways can you select one manufacturer, one car size, and one color? Use a tree
diagram to check your result.
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7. The access code for a car’s security system consists of four digits. Each digit can be 0 through 9.
How many access codes are possible if:
a.) each digit can be used only once and not repeated.
b.) each digit can be repeated?
c.) each digit can be repeated but the first digit cannot be 0 or 1?
8. You are making license plates for the State of Missouri.
a.) How many license plates can you make with 6 characters if you can only use letters?
b.) How many license plates can you make with 6 characters if you can use letter and numbers?
c.) How many license plates can you make with 6 characters (letters/#s) if you cannot repeat any
character?
d.) How many license plates can you make with 6 characters, if you must use 3 letters, 3 numbers, and
cannot repeat any letter or number?
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12.1b: The Complement Rule
Objectives:

Distinguish among classical probability and empirical probability.

Determine the probability of the complement of an event.
Notes:
Types of Probability: There are two types of probability
1)
2)
The probability that an event E will occur is written as P(E) and is read “the probability of event E.”
Notes: Classical (theoretical) Probability
Each outcome in a sample space is equally likely.
Classical Probability
P(E) =
Example: Theoretical Probability - You roll a six-sided die. Find the probability of each event.
1) Event A: rolling a 3
Event A = {
2) Event B: rolling a 7
Event B =
3) Event C: rolling a number less than 5.
Event C =
}
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Notes: Empirical (statistical) Probability


Based on observations obtained from probability experiments.
Relative frequency of an event.
Empirical Probability
P(E) =
When an experiment is repeated many times, regular patterns are formed. These patterns make it
possible to find
.
Example: Empirical Probability - A company is conducting an online survey of randomly selected
individuals to determine if traffic congestion is a problem in their community. So far, 320 people have
responded to the survey. What is the probability that the next person that responds to the survey says
that traffic congestion is a serious problem in their community?
Response
Number of times, f
Serious problem
123
Moderate problem
115
Not a problem
82
Σf =
Notes: The Law of Large Numbers
The Law of Large Numbers:
- Flip 10 coins, 100 coins, 1000 coins, 10000 coins in Excel to illustrate the phenomena.
Practice: Classify each statement as an example of classical or empirical probability.
1. The probability that you will be married by age 30 is 0.53.
2. The probability that you will draw a face card from a deck of cards in 3/13.
3. The probability that a voter chosen at random will vote Republican is 0.45
4. The probability of winning a 1000-ticket raffle with one ticket is 1/1000.
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Notes: Range of Probabilities Rule and Complement Rule
The probability of an event E is between 0 and 1, inclusive.
0 ≤ P(E) ≤ 1
Complement of event A:
Complement Rule
Denoted Ac (complement of A)
Practice: You survey a sample of employees at a company and record the age of each. Find the
probability of randomly choosing an employee who is not between 25 and 34 years old.
a) Use empirical probability to find P(age 25 – 34)
b) Use the complement rule
Employee
ages
Frequency, f
15 to 24
47
25 to 34
266
35 to 44
213
45 to 54
174
55 to 64
123
65 and over
32
Σf =
10
5. Suppose you know that the probability of getting the flu this winter is 0.43. What is the probability
that you will not get the flu?
6. Three coins are tossed in the air, what is the probability of getting at least one heads?
7. You throw 2 dice. What is the probability that the two scores are different?
8. A gumball machine contains gumballs of five different colors: 36 red, 44 white, 15 blue, 20 green,
and 5 orange. The machine dispenser randomly selects one gumball.
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11.2: Conditional Probability
Objectives:
• Determine conditional probabilities.
Notes: Conditional Probability
Conditional Probability:
Denoted P(B | A)
(read “probability of B, given A”)
Example: Conditional Probability
a) A card is selected from a standard deck. Find the probability that the card is a queen. Is this
conditional probability?
b) Two cards are selected in sequence from a standard deck. Find the probability that the second card
is a queen, given that the first card is a king. (Assume that the king is not replaced.) Is this conditional
probability?
c) Two cards are selected in sequence from a standard deck. Find the probability that
the two cards selected are a pair (the same).
d) You select a card from a deck. It is a face card. What is the probability that it is a Jack?
e) You select a card from a deck. It has a number on it. What is the probability that it is a 2?
f) You have selected 4 cards from a deck. They are all spades. What is the probability that the next card
you draw is a spade (no replacement)?
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Practice: Conditional Probability
1. The table shows the results of a study in which researchers examined a child’s IQ and the presence of
a specific gene in the child.
a) Find the probability that a child has a high IQ, given that the child has the gene. Is this conditional
probability?
Gene
Present
Gene not
present
Total
High IQ
33
19
52
Normal IQ
39
11
50
Total
72
30
102
b) Find the probability that the child does not have the gene. Is this conditional probability?
c) Find the probability that a child does not have the gene, given that the child has a normal IQ?
2. Conditional Probability - Pick a letter, any letter.
a) P(M)
b) P(E)
c) P(vowel)
d) P(consonant)
e) P(E|vowel)
f) P(vowel|E)
g) P(consonant|E)
h) P(T|consonant)
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The Monty Hall Problem: Suppose you're on a game show, and you're given the choice of three doors:
Behind one door is a car; behind the two other doors, goats. You pick a door, say No. 1, and the host,
who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to
you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Source: Parade Magazine
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12.3: The Multiplication Rule
Objectives:
•
Distinguish between independent and dependent events.
•
Use the Multiplication Rule to find the probability of two events occurring in sequence.
Notes: Independent and Dependent Events
In some events, one event does not affect the outcome of another. For example, if you roll a die and
flip a coin, the outcome of the roll of the die does not affect the probability of the coin landing on
heads. These two events are ______________.
Independent events
•
The occurrence of one of the events does not affect the probability of the occurrence of the
other event
•
Events that are not independent are ________________________
Practice: Decide whether the events and independent or dependent.
1. Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck
(B).
2. Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B).
3. E = The battery in your cell phone is dead. F = The batteries on your calculator are dead.
4. E = Your favorite color is blue. F = Your friend’s favorite hobby is fishing
5. A = You are late to school. B = Your car runs out of gas.
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Notes: The Multiplication Rule
•
The probability that two events A and B will occur in sequence is…
Multiplication Rule
For Dependent Events.…
For Independent Events…
- These can be extended for any number of events.
Examples: The Multiplication Rule
1. Two cards are selected, without replacing the first card, from a standard deck. Find the probability of
selecting a king and then selecting a queen.
2. A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.
Practice: Answer each of the following.
3. The probability that a particular knee surgery is successful is 0.85.
a) Find the probability that three knee surgeries are successful.
b) Find the probability that none of the three knee surgeries is successful.
c) Find the probability that at least one of the three knee surgeries is successful.
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4. More than 15,000 U.S. medical school seniors applied to residency programs in 2007. Of those, 93%
were matched to a residency position. 74% of the seniors matched to a residency position were
matched to one of their two top choices.
a. Find the probability that a randomly selected senior was matched a residency position and it was
one of the senior’s top two choices.
A={
}
B={
}
b. Find the probability that a randomly selected senior that was matched to a residency position did
not get matched with one of the senior’s top two choices.
5. Suppose that a satellite defense system is established in which four satellites acting independently
have a 0.9 probability of detecting an incoming ballistic missile. What is the probability that at least
one the four satellites detects an incoming ballistic missile? Would you feel safe with such a system?
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More Probability Practice! 220 random US adults were surveyed regarding marital status. The results
are below.
Males
31
Females
25
Married
63
64
Widowed
3
11
Divorced
10
13
Never Married
Totals
Totals
Practice: Use the table above to answer each of the following.
6. What is the probability that one is never married and widowed?
7. What is the probability that one is married and divorced?
8. What is the probability that one is male and Widowed?
9. What is the probability that married and male?
10. What is the probability that one is divorced and male?
11. What is the probability that one is divorced given they were male?
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12.4: The Addition Rule
Objectives:

Determine if two events are mutually exclusive.

Use the Addition Rule to find the probability of two events.
Notes: Mutually Exclusive
Mutually Exclusive:
A
B
A
B
Practice: Decide if the events are mutually exclusive.
1.
Event A: Roll a 3 on a die.
Event B: Roll a 4 on a die.
2.
Event A: Randomly select a male student.
Event B: Randomly select a nursing major.
3.
Event A: Randomly select a blood donor with type O.
Event B: Randomly select a female blood donor.
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Notes: Addition Rule
Addition rule for the probability of A or B
•
The probability that events A or B will occur is…
Addition Rule
For Events that are not Mutually Exclusive.…
For Mutually Exclusive Events…
*Can be extended to any number of mutually exclusive events.*
Practice: Answer each of the following. Sometimes it may help to draw a picture or diagram.
4. You select a card from a standard deck. Find the probability that the card is a 4 or an ace.
5. You roll a die. Find the probability of rolling a number less than 3 or rolling an odd number.
6. A die is rolled. Find the probability of rolling a 6 or an odd number.
7. A card is selected from a standard deck of cards. Find the probability that the card is a face card or a
heart.
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Let’s get this straight…

Two events E and F are _____________________if the occurrence of the event E in a
probability experiment does not affect the probability of event F. This applies to the
______________________ rule.

Two events A and B are ___________________________________ if they cannot occur at the
same time. This applies to the ____________________ rule.

The word and in probability implies that we use the ___________________ Rule.

The word or in probability implies that we use the ____________________ Rule.

True or False: When two events are mutually exclusive, they are also independent.
How about we look at it in terms of a Venn Diagram…
A
1
B
2
3
4
What regions correspond to each of the following…
Prob(A or B) =
Prob(A and B) =
Prob(A)c =
Prob(B)c =
Prob(A or Not B)
Prob(B or Not A)
Prob(Not A or Not B)
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8. A blood bank catalogs the types of blood given by donors during the last five days. A donor is
selected at random.
a) Find the probability the donor has type O or type A blood.
b) Find the probability the donor has type B or is Rh-negative.
9. In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an
A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an
A student?
10. A cooler contains 7 Pepsis, 5 iced tea, 9 waters, and 6 Sprites. If you reach in and randomly pull out
a drink, what is the probability that it is caffeinated or a soda?
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12.5a: Permutations
Objectives:

Determine the number of ways a group of objects can be arranged in order.

Determine the number of ways to choose several objects from a group without regard to order.

Use the counting principles to find probabilities.
Notes: Permutations
Permutation:
Permutations: The number of different permutations of n distinct objects is n! (n factorial)
n! =
0! =
Examples: Let’s look at a few examples.
Practice: Consider the following examples.
1. The teams in the National League Central Division are listed. How many different final standings are
possible?
National League Central Division
Chicago Cubs
Houston Astros
Pittsburgh Pirates
Cincinnati Reds
Milwaukee Brewers
St. Louis Cardinals
2. You have just been hired as a book representative for Prentice Hall. On your first day, you must
travel to seven schools to introduce yourself. How many different routes are possible.
23
3. Suppose Dan is going to make a spotify playlist that will contain 12 songs. In how many ways can
Dan arrange the 12 songs in the playlist?
Practice: Suppose you want to choose some of the objects in a group and put them order. Such an
ordering is called a permutation of n objects taken r at a time.
Permutation of n objects taken r at a time

The number of different permutations of n distinct objects taken r at a time.
Permutation Formula:
4. Find the number of permutations the letters ABC taken 2 at a time.
5. Find the number of ways of forming three-digit codes in which no digit is repeated.
6. How many ways can 7 horses finish to win in a race?
7. How many ways can 7 horses finish to show in a race?
Notes: Distinguishable Permutations
The number of distinguishable permutations of n objects where n1 are of one type, n2 are of another
type, and so on…
Distinguishable Permutations Formula:
24
Practice: Answer the following.
8. The letters AAAABBC can be rearranged 7! orders, but many of these are not distinguishable. The
number of distinguishable orders is
9. How many vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red.
10. A building contractor is planning to develop a subdivision that consists of 6 one-story houses, 4
two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be
arranged?
11. The contractor wants to plant six oak trees, nine maple trees, and five poplar trees along the
subdivision street. The trees are to be spaced evenly apart. In how many distinguishable ways can
they be placed?
12. The contractor wants to plant six oak trees, nine maple trees, and five poplar trees along the
subdivision street. The trees are to be spaced evenly apart. In how many distinguishable ways can
they be placed?
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12.5b: Combinations and Counting Principle Applications
Find the number of permutations the letters ABC taken 2 at a time.
Find the number of combinations of the letters ABC taken 2 at a time.
In each selection, order does not matter (AB is the same as BA). The number of ways to choose r
objects from n objects without regard to order is called the number of combinations of n objects taken
r at a time.
Combination of n objects taken r at a time

A selection of r objects from a group of n objects without regard to order
Combinations Formula:
Practice: Answer the following
1. A state’s department of transportation plans to develop a new section of interstate highway and
receives 16 bids for the project. The state plans to hire four of the bidding companies. How many
different combinations of four companies can be selected from the 16 bidding companies?
2. The manager of an accounting department wants to form a three-person advisory committee from
the 20 employees in the department. In how many ways can the manager form this committee?
3. How many different random samples of 5 can be obtained from a population whose size of 100.
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Practice: Answer the following questions. You will need to determine whether you are dealing with a
combination or permutation.
4. A student advisory board consists of 17 members. Three members serve as the board’s chair,
secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the
probability of selecting at random the three members that hold each position?
5. A student homecoming committee consists of 20 members. 4 members serve as the executive
board. Each member is equally likely to serve either of the positions. What is the probability of
selecting at random the four members that are on the executive board?
6. You have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly
arranged in order, what is the probability that the arrangement spells the word Mississippi?
7. 7 horses are in a race, you write down a guess of what order they will all come in. What is the
probability that you are correct?
8. You have 6 letters containing of one L, two E’s, two T’s, and one R. If the letters are randomly
arranged in order, what is the probability that the arrangement spells the word letter?
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12.6: Geometric Probability
1. A square is inscribed in a circle which is inscribed in another square.
A small item is tossed on this surface in such a way that it lands randomly at some point inside the
larger square…
a. What is the probability of landing on the interior of the circle?
b. What is the probability of landing inside the inner square?
c. What is the probability of landing inside the circle, but outside the inner square?
d. What is the probability of landing outside of the circle?
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2. A bull’s eye is below. You are not very good at throwing darts, so it is pretty random where your dart
lands. The diameter of the smallest circle is 3 cm, the diameter of the medium circle is 5 cm, the
diameter of the large circle is 7 cm.
Find the probability of…
a. Getting a bull’s eye?
b. Landing in the ring of the medium circle?
c. Landing in the ring of the outer circle?
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3. Two friends who take metro to their jobs from the same station arrive to the station uniformly randomly
between 7 and 7:20 in the morning. They are willing to wait for one another for 5 minutes, after which they take
a train whether together or alone. What is the probability of their meeting at the station?
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