Concept: 4.1 – What is probability?

I. What is probability?

a. Probability: ____________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

i.

ii. Fraction, Decimal, Percent

b. Notation

i. P(A) = “P of A” = probability of event A

1. If P(A) = 1, then _____________________________________________________

__________________________________________________________________

2. If P(A) = 0, then _____________________________________________________

__________________________________________________________________

c. Vocabulary

i. A statistical experiment or a statistical observation can be thought of as any random

activity that results in a definite outcome.

ii. An event is a ____________________________________________________________

_____________________________________of a statistical experiment or observation.

iii. A simple event is ________________________________________________________

_________________________________________________of a statistical experiment.

iv. The set of ___________________________________________________ makes up the

sample space of an experiment. The sample space can sometimes be listed.

Name: _______________________________

Date: ______________ Period: _______

AP/H Statistics

Triton Regional High School

Emily LeBlanc-Perrone

Unit A Guided Notes (Summer Work – Chapter 4: Elementary Probability Theory)

d. Probability Assignments

i. A probability assignment based on ___________________ incorporates past experience,

judgment, or opinion to __________________ the likelihood of an event. This is called

“subjective” probability.

ii. A probability assignment based on ____________________________________uses the

formula: 𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐞𝐯𝐞𝐧𝐭 = 𝐫𝐞𝐥𝐚𝐭𝐢𝐯𝐞 𝐟𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲 =

; where 𝑓 is the frequency

of the event occurrence in a sample of 𝑛 observations. This is called “experimental”

probability.

1. When you gather data from observations, you can calculate an experimental

probability. Each observation is an experiment or a trial.

a. 𝑷(𝑬) =𝐧𝐮 𝐛𝐞𝐫 𝐨𝐟 𝐭𝐢 𝐞 𝐭 𝐞 𝐞𝐯𝐞𝐧𝐭 𝐨𝐜𝐜𝐮𝐫

𝐧𝐮 𝐛𝐞𝐫 𝐨𝐟 𝐭𝐫𝐢𝐚𝐥

iii. A probability assignment based on ____________________________________uses the

formula: 𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐞𝐯𝐞𝐧𝐭 = 𝐮 𝐛𝐞𝐫 𝐨𝐟 𝐨𝐮𝐭𝐜𝐨 𝐞 𝐟𝐚𝐯𝐨𝐫𝐚𝐛𝐥𝐞 𝐭𝐨 𝐞𝐯𝐞𝐧𝐭

𝐨𝐭𝐚𝐥 𝐧𝐮 𝐛𝐞𝐫 𝐨𝐟 𝐨𝐮𝐭𝐜𝐨 𝐞 ; This is called

“theoretical” probability.

iv. Example: Consider each of the following events, and determine how the probability is

assigned.

1. A sports announcer claims that Sheila has a 90% chance of breaking the world

record in the 100-yard dash.

2. Henry figures that if he guesses on a true–false question, the probability of

getting it right is 0.50.

3. The Right to Health lobby claims that the probability of getting an erroneous

medical laboratory report is 0.40, based on a random sample of 200 laboratory

reports, of which 80 were erroneous.

v. How do you decide which method to use?

1. It depends on ______________________________________________________

__________________________________________________________________

2. The technique of using the relative frequency of an event as the probability of

that event is a common way of assigning probabilities and will be used a great

deal in later work.

e. Law of Large Numbers:

as the sample size increases, the relative frequencies (experimental probability) of the

outcomes get closer and closer to the theoretical (or actual) probability value

f. Example:

Professor Gutierrez is making up a final exam for a course in literature of the southwest. He

wants the last three questions to be of the true-false type. To guarantee that the answers do

not follow his favorite pattern, he lists all possible true-false combinations for the three

questions on slips of paper and then pick one at random from a hat.

i. List the outcomes of the sample space

ii. What is the probability that all three items will be false?

iii. What is the probability that exactly two items will be true?

II. Probability Facts

a. The sum of the probabilities of all simple events in a sample space _______________________

b. The complement of an event A is the event that _____________________________________,

denoted by Ac, or

i. P(A) + P(Ac) = 1

ii. P(event A does not occur) = P(Ac) = 1 – P(A)

c. Example:

The probability that a college student who has not received the flu shot will get the flu is 0.45.

What is the probability that a college student will not get the flu if the student has not had the

flu shot?

Summary

I think the Key Ideas are:

1. ___________________________________________________________________________________________

2. ___________________________________________________________________________________________

3. ___________________________________________________________________________________________

I learned:__________________________________________________________________________________________

__________________________________________________________________________________________________

Questions I have:

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

Concept: 4.2 (a) – Conditional Probability and Multiplication Rules

I. Vocabulary

a. A compound event consists of two or more simple events.

b. Two events are independent if the _________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

c. If events are dependent, _________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

Dependent Events Independent Events

Pick a card from a deck, then a second card

Roll a number cube, then spin a spinner

Choosing a ball from a bag, then choosing a second ball

Pick a card from a deck, put it back, then pick another card

Choosing a ball from a box, and a cube from a bag

d. If the events are dependent, then we must take into account the changes in the probability of

one event caused by the occurrence of the other event.

i. Conditional probability is the probability that __________________________________

________________________________________________________________________

________________________________________________________________________

1. Notation: ( | ) = “the probability of B, given A”

II. Some Probability Rules

a. The event A and B consists _______________________________________________________

______________________________________________________________________________

______________________________________________________________________________

i. This is called the intersection and uses the symbol .

b. Multiplication Rules

i. Independent Events: ( 𝑛 ) = ( ) = ( ) ( )

ii. Dependent Events:

( 𝑛 ) = ( ) = ( ) ( | ) or ( 𝑛 ) = ( ) = ( ) ( )

c. Conditional Probability ( | ) = ( )

( )

i. Two events A and B are said to be independent if ( | ) = ( )

d. Examples:

i. Suppose you are going to throw two fair dice. What is the probability of getting a 5 on

each die?

ii. A utility research company asked 50 of its customers whether that pay their bills online

or by mail. What is the probability that a customer pays bills online given that the

customer is male?

iii. Consider a collection of 6 balls that are identical except in color. There are 3 green balls,

2 blue balls, and 1 red ball. Compute the probability of drawing 2 green balls from the

collection if the first ball is not replaced before the second ball is drawn.

Summary

I think the Key Ideas are:

1. ___________________________________________________________________________________________

2. ___________________________________________________________________________________________

3. ___________________________________________________________________________________________

I learned:__________________________________________________________________________________________

__________________________________________________________________________________________________

Questions I have:

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

Concept: 4.2 (b) – Addition Rules

I. Vocabulary

a. Two events are mutually exclusive or disjoint _______________________________________.

In particular, events A and B are mutually exclusive if P(A and B) = 0.

II. More Probability Rules

a. The condition A or B is satisfied by any of the following:

i. Any outcome in A occurs

ii. Any outcome in B occurs

iii. Any outcome in both A and B occurs

This is called the union and uses the symbol .

b. Addition Rules

i. Mutually Exclusive Events: ( ) = ( ) = ( ) ( )

ii. Non-Mutually Exclusive Events: ( ) = ( ) = ( ) ( ) ( 𝑛 )

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=kgUu6yp3kEEH2M&tbnid=eKVe5IznqxsAvM:&ved=0CAUQjRw&url=http://www.emathzone.com/tutorials/basic-statistics/mutually-exclusive-events.html&ei=-SmvUdb2B4364AOktIH4Ag&bvm=bv.47380653,d.dmg&psig=AFQjCNF3j3VXF64Kx018MeHxL52w0_N6OA&ust=1370520433077863

c. Examples

i. Consider an introductory statistics class with 31 students. The students range from

freshmen through seniors. Some students are male and some are female. Figure 4-5

shows the sample space.

1. Suppose we select one student at random from the class. Find the probability

that the student is either a freshman or a sophomore.

2. Select one student at random from the class. What is the probability that the

student is either a male or a sophomore?

ii. Laura is playing Monopoly. On her next move she needs to throw a sum bigger than 8

on the two dice in order to land on her own property and pass Go. What is the

probability that Laura will roll a sum bigger than 8?

Summary

I think the Key Ideas are:

1. ___________________________________________________________________________________________

2. ___________________________________________________________________________________________

3. ___________________________________________________________________________________________

I learned:__________________________________________________________________________________________

__________________________________________________________________________________________________

Questions I have:

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

Concept: 4.3 – Trees and Counting Techniques

I. Counting Techniques

a. Counting Techniques help _______________________________________________________

in larger sample spaces or those formed by more complicated events.

II. Fundamental Counting Principle (Multiplication Rule of Counting): gives the total number of possible

outcomes for a sequence of events.

a. It is the _______________________________________________________________________

______________________________________________________________________________

b. Example:

Jacqueline is in a nursing program and is required to take a course in psychology and one in physiology

(A and P) next semester. She also wants to take Spanish II. If there are two sections of psychology, two

of A and P, and three of Spanish II, how many different class schedules can Jacqueline choose from?

(Assume that the times of the sections do not conflict.)

III. Tree Diagrams

a. A tree diagram gives a visual display of the total number of outcomes of an experiment

consisting of a series of events.

i. The number of end branches gives the total number of outcomes

ii. Displays the individual outcomes

iii. Helps compute the probability of an outcome displayed in the tree

b. Example:

Jacqueline is in a nursing program and is required to take a course in psychology and one in

physiology (A and P) next semester. She also wants to take Spanish II. There are two sections of

psychology, two of A and P, and three of Spanish II.

Make a tree diagram that shows all the possible course schedules for Jacqueline.

c. Example:

Suppose there are 5 balls in an urn. They are identical except for color. 3 balls are red and 2 are

blue. You are instructed to draw out one ball, note its color, and set it aside. Then you are to

draw out another ball and note its color. What are the outcomes of the experiment? What is

the probability of each out come?

IV. Permutation

a. Permutations allow us to compute the number of different ordered arrangements, when we use some

of the items being arranged.

b. The Permutation Rule:__________________________________________________________________

_______________________________________________________________ is = =

( )

where 𝑛 and are whole numbers and 𝑛 .

i. Note that there are two commonly used notations

c. Examples

i. Compute the number of possible ordered seating arrangements for eight people in five

chairs.

ii. The board of directors at the Belford Community hospital has 12 members. Three

officers hold the following positions: president, vice president, and treasurer. How many

ways can the board elect members to these positions?

V. Combination

a. Combinations allow us to compute the number of different groupings or combinations of items. Note

that the order does NOT matter!

b. The Combination Rule: _________________________________________________________________

_____________________________________________________________ = = (𝑛 ) =

( )

where 𝑛 and are whole numbers and 𝑛 .

i. Note that there are three commonly used notations

c. Examples

i. In your political science class, you are assigned to read any 4 books from a list of 10

books. How many different groups of 4 are available from the list of 10?

ii. Three of the 12 members of the board of directors at the Belford Community hospital

will be chosen to attend a conference, all expenses paid. How many how ways can the

board choose a group to attend the conference?

Summary

I think the Key Ideas are:

1. ___________________________________________________________________________________________

2. ___________________________________________________________________________________________

3. ___________________________________________________________________________________________

I learned:__________________________________________________________________________________________

__________________________________________________________________________________________________

Questions I have:

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________