Download of the Sample Space

Unit 9 - Textbook Notes
HSS - CP.A.1 -#1 Geometry/Geometry Honors
Sample Spaces as Sets
Probability is often defined as how likely something is to happen. In determining how likely something is to
happen we must first determine what the total number of things that could happen for any given event or
experiment. The list of all possible outcomes is called the sample space.
A LIST -- A sample space is often organized into a set of elements. For example the sample space for rolling a
die is {1, 2, 3, 4, 5, 6} or the sample space for flipping a coin is {Head, Tail}. These two sample spaces represent
uniform probability. Uniform probability is where each element of the set has the same chance of happening
or in other words, each element is equally likely to happen. In a uniform probability, the number of elements
in the set is the total number of outcomes possible for the sample space.
Some sample spaces are not uniform, such as a bag of marbles with 2 red and 1 green, the sample space is
{red, green} because only a red or a green marble can be chosen. But in this sample space each element is
NOT equally likely to happen because there are 2 ways to pick a red marble and only 1 way to pick a green.
While there are only two elements listed in the sample space, the total number of outcomes of this sample
space is 3, written n(S) = 3, because red has 2 ways of obtaining it and green has 1. The notation n(S) refers to
the total number of outcomes possible in the set. Also notice when we list the elements of a set we use the
set brackets {} and each element in the set is separated by a comma. {12, 34} is a set containing only two
elements, whereas the set {1, 2, 3, 4} is a set containing 4 elements.
Flipping a Coin
Rolling a Six Sided Dice
Spinning a 4 color spinner
Rolling a Weighted Dice
SAMPLE SPACE
{Head, Tail}
SAMPLE SPACE
{1, 2, 3, 4, 5, 6}
SAMPLE SPACE
{Red, Yellow, Green, Blue}
SAMPLE SPACE
{4, 5, 6}
Uniform
Uniform
Uniform
Not Uniform
Picking a flavor of ice cream
Determining the gender of baby
Picking from a bag of marbles
SAMPLE SPACE
{Chocolate, Vanilla, Strawberry}
SAMPLE SPACE
{Boy, Girl}
SAMPLE SPACE
(Blue, Red}
Uniform
Uniform
Not Uniform
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Some typical sample spaces written in set notation are:
Unit 9 - Textbook Notes
Some sample spaces can be quite large and difficult to list such as the 52 cards in a standard deck.
Large sample spaces can make listing them a daunting task and often quite inefficient. For this reason we
have a few other ways to organize a sample space.
A TREE DIAGRAM. The first tree diagram below shows two flips of a coin. The first flip could be a head or a
tail and then the second flip could produce a head or a tail from each of the previous
options. The tree displays the sample space for flipping a coin twice as {HH, HT, TH, TT}.
We also notice the number of elements of the sample space is (2)(2) = 4. In the second
tree diagram below we see the sample space for selecting two scoops of ice cream where
the choices are vanilla, chocolate, and strawberry. There are three choices for the first scoop and then the
second choice gives us three more options for each previous option. The tree diagram displays the sample
space for choosing a double scoop as {VV, VC, VS, CV, CC, CS, SV, SC, SS}. Again we notice an easy method for
determining the number of elements in this sample space is to multiply 3 by 3 to get 9.
Randomly Selecting a Two Scoops Ice Cream Cone
HEAD HH
HEAD
VANILLA
TAIL
HEAD
HT
TH
CHOCOLATE
TAIL
1ST FLIP
(2)
TAIL
2ND FLIP
(2)
TT
TOTAL
(2)(2) = 4
STRAWBERRY
3 OPTIONS
In this case, getting a head is as likely as
getting a tail. Thus this tree diagram
represents a uniform probability. In this
sample space each event is equally likely –
getting a {Tail, Tail} is as likely as getting a
{Head, Tail} or any of the other events.
VANILLA
VV
CHOCOLATE
VC
STRAWBERRY
VS
VANILLA
CV
CHOCOLATE
CC
STRAWBERRY
CS
VANILLA
SV
CHOCOLATE
SC
STRAWBERRY
SS
3 OPTIONS
(3)(3) = 9
OPTIONS
In this case, the choice of flavors might not be equally likely.
If chocolate was the most popular flavor it would have a
greater chance of being selected. The fact that each choice
would not be equally likely would make the probabilities
non-uniform. For example, getting {Chocolate, Chocolate}
would be more likely to occur than {Strawberry, Strawberry}.
Page125
Tree Diagram for Flipping a Coin Twice
Unit 9 - Textbook Notes
A TABLE OR CHART. A tree diagram would probably not be the best way to organize the sample space for all
possible outcomes when rolling two dice. The tree would have 6 initial branches and then for each of those
branches there would 6 more branches – in just two levels of the tree we would already have 36 branches.
This would create a very large (hard to draw) tree diagram. A better way to organize this data might be to
create a table.
1
2
3
4
5
6
1
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
2
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
3
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
4
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
5
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
6
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
When dealing with the occurrence of more than one event or activity such as this one, it is important to be
able to quickly determine how many possible outcomes exist without listing all of the possible events. In this
case to determine the total number of outcomes we could simply multiply 6 times 6 to get 36. This simple
multiplication process is known as the Fundamental Counting Principle.
HSS - CP.A.1 -#2 Geometry/Geometry Honors
It is important to be able to quickly determine how many possible outcomes exist without listing them all.
Listing outcomes can be time consuming and in a large set it is easy to miss some. For example in the case of
rolling a die twice in a previous objective we used a table to find that there were 36 different outcomes that
could happen but we notice that we could have simply multiplied 6 times 6 to get the 36 value instead of
listing them all out. This simple multiplication process is known as the Fundamental Counting Principle.
FUNDAMENTAL COUNTING PRINCIPLE
If there are a ways for one activity or event, and b ways for a second event to occur, then there are a
•b
ways for both to occur.
Ex. #1 - The Club Restaurant offers 6 appetizers, and 14 main courses. What
is the sample space if a person orders one appetizer & one main course?
(6)(14) = 84
Ex. #2 – A coin if flipped, a dice is rolled and a card is picked. What is the
sample space for these events?
(2)(6)(52) = 624
Ex. #3 - The pass lock on a iPad requires a 4 digit password. What is the
sample space for all 4 digit passcodes?
(10)(10)(10)(10) = 10,000
Ex. #4 - Jeff has 5 t-shirts, 3 pants, and 4 pairs of shoes. What is the sample
space for all possible outfits?
(5)(3)(4) = 60
The fundamental counting principle handles the calculation for most sample spaces. We will look at the use of
permutations and combinations to handle other types of situations at a later time.
Page126
The number of ways in which a series of successive things can occur
is found by multiplying the number of ways in which each thing can occur.
Unit 9 - Textbook Notes
HSS - CP.A.1 -#3 Geometry/Geometry Honors
Intuitive Idea of Probability
We use ratios to show how likely, or unlikely, an outcome might be. This ratio is called the probability of the
event. A probability is expressed as:
The numerator is the number of elements in the sample space that meet the
desired criteria. n(A) reads the “number of” outcomes of event A.
The denominator is the total number of possible outcomes in the sample
space. n(S) reads the “number of” outcomes in the sample space S.
1
. The probability of getting a red marble from a
2
3
1
bag of 3 red and 2 blue marbles is, . The probability of getting a five on a single roll of a die is, .
5
6
The probability of getting a head on a single flip of a coin is,
We use notation P(A), to represent the probability of event A.
To notate the previous examples it would look like, P(Head) =
P( A) 
n( A)
n( S )
1
3
1
, P(Red) = and P(5) = ,
2
5
6
Ratios are most often expressed in the form of a fraction. Fractions, however, can be expressed as decimals or
percent. So probabilities may be expressed as fractions, decimals or percent.
The range of the values for a probability is from 0 to 1. To have a probability of 0 means that it is impossible
for the event to happen such as picking a red marble from a bag of 7 blue and 3 white marbles. To have a
probability of 1 means that it is a certain event such as picking an even numbered tile from a bag that contain
tiles with the numbers 2, 4, 6 and 12.
As the chart on the left shows,
probabilities range from 0 to 1.
If an event is impossible and will never occur,
the probability is 0.
An eight sided die is rolled, and the outcome noted. Determine whether each of the following outcomes is:
certain to happen, certain not to happen, likely to happen, unlikely to happen.
1
P(Getting a 3) – The probability is which is UNLIKELY TO HAPPEN.
8
5
P(Getting a number greater than 3) – The probability is which is LIKELY TO HAPPEN.
8
8
P(Getting a value less than 10) – The probability is which is CERTAIN TO HAPPEN.
8
0
P(Getting a 12) – The probability is which is CERTAIN NOT TO HAPPEN.
8
Page127
If an event is absolutely certain to occur,
the probability is 1.
Unit 9 - Textbook Notes
Determine if the events are: (I)mpossible, (UN)likely, (EQ)ually likely, (L)ikely or (C)ertain to happen.
a) Getting red (R) on the 4 color
spinner.
b) Rolling this dice and getting a
value greater than 1.
UNLIKELY
LIKELY
c) Selecting a football from these
balls.
IMPOSSIBLE
Determine the basic probability.
In a bag of 1 red, 5 green and 4 yellow marbles picking a….
P (Yellow) =
4
10
P (Colored Marble) =
10
10
P (Green) =
5
10
P (A Jack) =
4
52
In a standard deck of cards picking a…..
P (Spade) =
13
52
P (8 of hearts) =
1
52
Rolling this special die once….
P (3) =
2
6
P (4) =
1
6
P (Not getting a 2) =
4
6
HSS - CP.A.1 -#4, #5 & #6 Geometry/Geometry Honors
A sample space represents all things that could occur for a given event. In set theory language this would be
known as the Universal Set – all elements defined by that set. We often use Venn diagrams to display the
relationships within sets and sample spaces.
Diagramming the Universal Set (Sample Space)
The universal set is usually diagrammed as a rectangle. The set name which is being used as the universal set
is usually placed in the upper left hand corner of the shape. Depending on the size of the set you do not have
to include all elements of the set in the diagram, usually a few are provided to give an image of some of the
values of the set. If the set is small, then all elements should be listed.
To diagram our sample space, the set M, a bag of
marbles with 4 red marbles (solid) and 6 white
marbles (empty) we create the rectangle, label it the
universal set M, and then list out the elements of the
set. In this case because there are only 10 elements
it is easy to list them all out in the diagram.
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Diagramming the Sample Space using Venn Diagrams
Unit 9 - Textbook Notes
Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
As stated earlier, a probability has two components, the sample space, which
represents all possible things that could happen, and the defined successful
outcomes, which represents the number of times a particular event occurs in that
sample space. The outcome could be picking a heart from a deck of cards, rolling an
even number on a dice, spinning a spinner and getting blue….. an outcome is simply
a subset of the universal set. A subset is a collection of elements that all exist
within another set. If all elements of set X belong to set Y, then it is said that set X is a subset of set Y. Any set
formed with elements of the universal set is a subset of that universal set.
For example if the sample space was rolling a D12 (a 12 sided dice) some subsets might be:
Rolling a prime number,
Set P = {2, 3, 5, 7, 11} is a subset of
Set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Rolling an even number less than 5,
Set E = {2, 4} is a subset of
Set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Rolling a number greater than 12,
Set B = {} is a subset of
Set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
In the third example, Set B is an EMPTY SET or NULL SET. This means that no elements fit that description.
The empty set gets its own special symbol, Ø. When notating an empty set we would write Set B = Ø, and NOT
Set B = {Ø}. The latter notation is wrong because that set contains one element, the empty set. Thus set B
would not be empty if it has one element, even if that element represents a set that has nothing.
When writing that one set is a subset of another we use two special mathematical symbols, either  or  .
The first symbol,  , allows the subset to be the same as or smaller, whereas the second symbol,  , forces
the subset to contain less elements than the original set and these subsets are called proper subsets. Now if
we look back to examples #1, #2 and #3, we would write those relationships as:
Ex. #1 Set P  Set U
or Set P  Set U
Ex. #2 Set E  Set U
or Set E  Set U
Ex. #3 Set B  Set U or   Set U
or Set B  Set U or   Set U
This can easily be turned into a probability - What is the probability of
picking a red marble from this bag of marbles?
n( R )
4

P(Set R) = P(Red) =
or 0.4 or 40%
n( S ) 10
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Diagramming a Outcome (Subset) using a Venn Diagram
When a subset is defined, the elements are organized and a new boundary
is drawn in the Venn diagram. So if we defined the set R as the set of all
red marbles in the bag we would draw a new boundary that would contain
all of those elements.
Set R = {3R, 4R, 5R, 8R}
Set U = {3R, 4R, 5R, 8R, 1W, 2W, 4W, 5W, 8W, 9W}
Set R  Set U
Unit 9 - Textbook Notes
If we defined set E to be the set of all even numbers in the bag we could
determine the probability to be:
Set E = {4R, 8R, 2W, 4W, 8W}
Set E  Set U
n( E )
5

P(Set E) = P(Evens) =
or 0.5 or 50%
n( S ) 10
Again if we defined set L to be the set of all numbers greater than 3 in the
bag, we could determine the probability to be:
Set L = {4R, 5R, 8R, 4W, 5W, 8W, 9W}
Set L  Set U
n( L )
7

P(Set L) = P(Numbers >3) =
or 0.7 or 70%
n( S ) 10
The Complement of an Event, “not”
The complement of an event is the probability of everything but that event occurring. So if the event was
set A, then the complement is denoted as, set Ac, everything that A is not. If the probability of picking a
3
8
, then its complement, the probability of not yellow is . An easy way to
11
11
c
calculate the complement is P(A ) = 1 – P(A). This works because all probabilities sum to 1 and so whatever
the probability of event A happening is, the probability of it not happening is everything else or in other words,
1 – P(A). This relationship is easily viewed in a Venn diagram.
yellow marble from a bag is
P(A) + P(Ac) = 1
When determining the probability of a complement it is usually simplest to calculate the probability of the
event and then subtract it from 1.
Ex. #1 – Given a bag of marbles with 3 green, 2 yellow and 5 red. What is the probability of NOT getting a
green marble?
3
10
1 – P(G) = P(Gc) 1 -
3
7
=
10 10
Page130
P(G) =
Unit 9 - Textbook Notes
Ex. #2 – When rolling a single die, what is the probability of NOT getting a 6?
P(A) =
1
6
1 – P(A) = P(Ac) 1 -
1 5
=
6 6
Ex. #3 – When picking a card from a standard deck, what is the
probability of NOT getting a diamond?
1 – P(A) = P(Ac) 1 -
13 39
=
52 52
Mutually Exclusive or Disjoint Sets
More than one subset can be defined at a time from a universal set, so for example we could define the set of
all red marbles, or the set of all even numbers, or the set of red marbles with numbers greater than 3 - the list
seems like it could go on forever. Sometimes when we define more than one set at a time they have no
elements in common. This is known as being mutually exclusive or disjoint. Two events are mutually
exclusive events if the events cannot both occur in the same trial of an experiment, for example the flip of a
coin cannot be both heads and tails and thus those two events are mutually exclusive.
Diagramming Disjoint Sets
If we define the set R to be all red marbles and the set W to
be all white marbles we get two mutually exclusive sets
because they have no elements in common with each other.
We diagram this relationship by drawing boundaries around
each set so that they do not touch or overlap in anyway.
Set R = {3R, 4R, 5R, 8R} and Set W = {1W, 2W, 4W, 5W, 8W, 9W}
Another example of disjoint sets would be set E, all of the even marbles, and set
O, all of the odd marbles.
Set E = {4R, 8R, 2W, 4W, 8W} and
Set O = {3R, 5R, 1W, 5W, 9W}
Page131
In both of these cases you cannot be both red and white or even and odd, thus they are mutually exclusive.
Unit 9 - Textbook Notes
The Intersection, “AND”
Of course when we define more than one subset the sets are not always mutually
exclusive. Sometimes the two sets have shared or common elements in them. The shared
items or elements are called the intersection of the sets. This should make sense to a
Geometry or Algebra I student because we have already discussed the intersection of two
lines. The intersection of two lines is a point, the only thing they HAVE IN COMMON.
The Intersection
The intersection is the collection of elements that are COMMON between the sets. The symbol notation for
intersection is  . In general, for any two sets S and T, the set consisting of the elements belonging to BOTH
set S and set T is called the intersection of sets S and T, denoted by Set S  Set T. This is sometimes also
described as the elements that are in set S AND in set T.
An example of two sets that would have an intersection could be found easily in a standard deck of cards, the
set R, all red cards, and the set Q, the set of all queens. These two sets are NOT mutually exclusive because
these sets would share two elements, the queen of hearts and the queen of diamonds. These two cards are
the intersection because they are in set R AND in set Q. Another example of an intersection in a deck of cards
would be the set D, the diamonds, and the set F, the face cards. The cards that are in set D AND set F (the
intersection) are the jack, queen, and king of diamonds.
Diagramming the Intersection
If we define the set R to be all red marbles and the set E to
be all even numbered marbles we get two sets that have an
intersection. When these two set get diagrammed they
have an overlapping region, a region that represents the
values that are in both sets. We usually shade that region.
Set R = {3R, 4R, 5R, 8R}
and
Set E = {4R, 8R, 2W, 4W, 8W}
Set R  Set E (Set R AND Set E) = {4R, 8R}
Another example of an intersection would be the set D, all numbers divisible
by 3, and the set W, all the white marbles.
Set D = {3R, 9W}
and
Set W = {1W, 2W, 4W, 5W, 8W, 9W}
Set D  Set W (Set D AND Set W) = {9W}
Could the intersection of two sets be empty? Of course if the two sets are mutually exclusive then there will
be no elements in the intersection of the two sets. For example,
the set E, the even numbered marbles and set O, the odd numbered marbles,
will have no elements in common and so the intersection is the empty set.
Set E = {4R, 8R, 2W, 4W, 8W} and Set O = {3R, 5R, 1W, 5W, 9W}
Page132
Set E  Set O (Set E AND Set O) = 
Unit 9 - Textbook Notes
The Union, “OR”
The union of sets is exactly what it sounds to be, the process of combining sets together to form a larger set.
The union of sets is the collection of all elements from both sets. The symbol for union is  (this is easier to
remember  nion). In general, for any two sets S and T, the set consisting of all the elements belonging to at
least one of the sets S and T is called the union of S and T, denoted Set S  Set T. This is sometimes also
described as the elements that are in set S OR in set T.
An example of a union could be found easily in a standard deck of cards, the set R, all red cards, and the set S,
the set of all spades. The union of these two sets would include all the hearts, all the diamonds and all the
spades. These cards are the union because it contains set R OR set S.
Diagramming the Union
Usually we don’t change the boundaries of the original sets to represent the
new union; usually we simply shade in the sets that have formed the new
union. The example to the right demonstrates the union of two mutually
exclusive sets, set W, the white marbles {1W, 2W, 4W, 5W, 8W, 9W} and set E,
the even red marbles {4R, 8R}.
Set W  Set E (Set W OR Set E)
{1W, 2W, 4W, 5W, 8W, 9W}  {4R, 8R}
= {1W, 2W, 4W, 5W, 8W, 9W, 4R, 8R}
An example of a union when the two sets that
would have an intersection would be the Set E,
the even numbers {2W, 4W, 8W, 4R, 8R} and the
set R, the red marbles {3R, 4R, 5R, 8R} .
Set E  Set R (Set E OR Set R)
{2W, 4W, 8W, 4R, 8R}  {3R, 4R, 5R, 8R}
= {2W, 4W, 8W, 3R, 4R, 5R, 8R}
You do not double list
elements in the set.
Let me do another example, the set B, the
marbles greater than 2 and the set T, the marbles
with a 3 or 4.
Set B  Set T (Set B OR Set T)
= {3R, 4R, 5R, 8R, 4W, 5W, 8W, 9W}
You do not double list
elements in the set.
Page133
{3R, 4R, 5R, 8R, 4W, 5W, 8W, 9W} 
{5R, 8R, 5W, 8W, 9W}
Unit 9 - Textbook Notes
HSS - CP.A.2 -#1 Geometry/Geometry Honors
Independence
When working with probabilities we often perform more than one event in a sequence this is called a
compound probability. Compound probabilities are more complex than a single event probability to compute
because the first event might affect the probability of the second event happening.
For example, the probability of getting a head on a single flip of a coin is ½. If you flip the coin
and get a head, the second flip’s probability of getting a head is still ½ because the results of
the first flip does not in any way affect the second flip. The second flip has the exact same
probability as if it was the first flip, ½. When the first action does not affect the second
action’s probability in any way the events are known to be INDEPENDENT.
In contrast, if you have a jar of cookies with 7 chocolate chip cookies and 3 peanut
butter cookies the probability of getting a chocolate chip is 7/10 and the probability of
getting a peanut butter cookie is 3/10. Is the probability of getting a peanut butter
cookie still 3/10, if you first pick out a chocolate chip cookie and eat it? Of course not,
the probability of getting a peanut butter cookie now is 3/9 because a chocolate chip
cookie is gone from the jar. The second selection is affected by the first selection, thus these two events are
NOT INDEPENDENT.
Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B
occurring (or vice versa).
Some other examples of independent events are:
-- Getting a head after tossing a coin AND selecting a purple marble from a bag.
-- Getting on head after tossing a coin AND rolling a 2 on a single 20-sided die.
-- Choosing a jack from a deck of cards, replacing it, AND then choosing a king as the second card.
What is the probability of rolling a 6 AND then getting a head on a coin flip?
These two events are independent - the die could roll any number of times and it would in no way influence
the flip of the coin. Let us use a set list, a tree and a Venn diagram to understand this problem.
Set S = Rolling a 6 = {6H, 6T}
Set H = Heads = {1H, 2H, 3H, 4H, 5H, 6H}
Set S  Set H = {6H}
1
2
3
P(6 AND H) = 1 / 12
Remember that AND represents the intersection.
4
5
6
H
1H
T
1T
H
2H
T
H
T
2T
3H
3T
H
T
H
4H
4T
5H
T
5T
H
6H P(6H) =
T
6T
1
6
x
1
2
=
1
12
Page134
Set U = {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}
Unit 9 - Textbook Notes
To find the probability of two independent events that occur in sequence, find the probability of each event
occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below.
Note that multiplication is represented by AND.
When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)
Example #1
What is the probability of rolling a 6 and
then rolling a 5?
1
1
P(S) =
P(F) =
6
6
 1  1  1
P(S and F) = P(S)  P(F) =    
 6  6  36
Example #3
What is the probability of getting a head on a coin
flip and then choosing a purple marble from a bag
that has 2 purple, 1 green, and 2 orange marbles?
These two events are independent. Let us use a tree
diagram to understand and solve this problem. What
is the P (Head and Purple)?
2
5
1
2
1
HEAD
5
2
5
2
5
1
2
1
TAIL
5
2
PURPLE
P(HP) =
1
2
x
2
5
=
Example #2
Given a bag of marbles with 3 red, 2 green and 5
yellow. What is the probability of choosing a red,
replacing it, and then choosing a green?
3
2
P(R) =
P(G) =
10
10
6
 3  2 
P(R and G) = P(R)  P(G) =    
 10  10  100
Example #4
A true and false question is followed multiple choice
question with possible four answers (1 correct & 3
wrong). What is the probability of getting both
questions correct, P(CC)?
1 1
1
1 COORECT
P(CC) = x =
2
4
8
MC
4
2
10
1
GREEN
2
CORRECT
T/F
3 WRONG
MC
4
ORANGE
PURPLE
1 WRONG
T/F
2
GREEN
1 COORECT
MC
4
3 WRONG
MC
4
ORANGE
5
Understanding independence is critical to probability because we must always take into account how one
event affects the next event.
The Date
The Amount of Sunlight
Not Independent
Income
Event #2
Education Level
Not Independent
c) Event #1
Event #2
Your Height
Your Favorite Color
Independent
Determine the P (A and B) given that Event A and Event B are independent.
a) P(A) = 0.1, P(B) = 0.3
b) P(A) = 0..25, P(B) = 0.8
c) P(A) = 0.5, P(B) = 0.5
P(A and B) = 0.03
P(A and B) = 0.2
P(A and B) = 0.25
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Determine if the following are independent or not.
a) Event #1
Event #2
b) Event #1
Unit 9 - Textbook Notes
Mutually Exclusive and Independence
A common misunderstanding is that independence is the same thing as being mutually exclusive. I get why
this is confusing, to be independent in a typical English language context means to be alone or separate which
is basically what we understand mutually exclusive to mean. This definition of independence is NOT the
mathematical one. Independence is about whether one event affects another event’s probability or not.
Mutually exclusive sets are those that don’t share any elements and independent sets are those that don’t
impact each other’s probabilities.
Mutually exclusive is about the sharing of elements,
and independence, is about affecting each other.
In the example to the right, event A and event B are NOT MUTUALLY
EXCLUSIVE because there is an intersection between the two sets.
But Events A and Event B are INDEPENDENT because
P(A)  P(B) = (0.3)(0.5) = 0.15
P(A AND B) = 0.15
So here is one example where mutually exclusive and independence
obviously are two different things!!
Replacement and No Replacement
The terms replacement and no replacement get used a lot in compound probabilities problems because they
describe what you did with the first thing that you selected… did you put it back or did you keep it?
P (Getting a green marble, replacing it, and getting a green marble)
Independent
P (Picking a black queen, not replacing it, and getting an ace)
Not Independent
These two words are HUGE clues as to whether the events are going to be independent or not.
REPLACEMENT
Because the item is replaced, it resets the event back to the original arrangement and
no probabilities are altered. Thus REPLACEMENT tells us that the events are
INDEPENDENT.
NO REPLACEMENT
Because the item is NOT replaced, the probabilities are altered. Thus NO
REPLACEMENT tells us that the events are NOT INDEPENDENT.
Determine if the following are independent or not.
Independent
b) picking a marble from a bag and
then spinning a 4 color spinner.
Independent
c) picking a card from a standard
deck, not replacing the card and
then picking again.
Not Independent
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a) picking a marble from a bag,
replacing the marble and then
picking again.
Unit 9 - Textbook Notes
Testing for Independence
We can use the formal relationship of P (A and B) = P(A)  P(B) to test independence. Here are three examples
of how we could use probabilities to determine if they are independent of each other.
Example #1
P(A) = 0.8
P(B) = 0.4
P(A and B) = 0.2
These are Not Independent because P(A)  P(B) = (0.8)(0.4) = 0.32 and this is not
the same as P(A and B) = 0.2 provided.
Example #2
P(A) = 0.6
P(B) = 0.5
P(A and B) = 0.3
These are Independent because P(A)  P(B) = (0.6)(0.5) = 0.30 and this is
the same as P(A and B) = 0.3 provided.
The same kind of test can occur but the information can be given in Venn diagram form.
Example #3
Check to see if P(A)  P(B) = P (A and B)
P(A) = 0.5
P(A)  P(B) = (0.5)(0.5) = 0.25
P(B) = 0.5
P (A and B) = 0.25
P(A AND B) = 0.25
INDEPENDENT
Example #4
Check to see if P(A)  P(B) = P (A and B)
P(A) = 0.7
P(A)  P(B) = (0.7)(0.2) = 0.14
P(B) = 0.2
P (A and B) = 0.14
P(A AND B) = 0.14
INDEPENDENT
Example #5
Check to see if P(A)  P(B) = P (A and B)
P(A) = 0.50
P(A)  P(B) = (0.5)(0.28) = 0.14
P(B) = 0.28
P (A and B) = 0.2
P(A AND B) = 0.20
NOT INDEPENDENT
Determine if the following are independent or not.
a)
b)
c)
P(A) = 0.45
P(B) = 0.6
P (A and B) = 0.27
(0.25)(0.6) = 0.15
Independent
(0.45)(0.6) = 0.27
Independent
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(0.3)(0.3) = 0.9
Not Independent