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SECTION 15.4 PROBABILITY SPACES
EVENT: any subset of the sample space or any set of individual outcomes
 Simple Event: an event of exactly one outcome

Impossible Event: an event with no outcomes;

Certain Event: an event with all outcomes of the sample space;
EXAMPLE #1: Consider the random experiment of tossing a coin three times.
SAMPLE SPACE:
E1: Toss 2 or more heads.
E2: Toss more than 2 heads.
E3: Toss 2 or less heads.
E4: Toss no heads.
E5: Toss Exactly 1 tail.
E6: First Toss is heads.
E7: Same number of heads as tails
E8: Toss 3 heads or Less
EXAMPLE #2: Consider the random experiment of tossing a coin and rolling a die.
SAMPLE SPACE:
E1: An even number.
E2: A head.
E3: A head and an odd number.
E4: A tail and prime number.
E5: A number less than 7.
E6: A number less than 3
E7: A 7 is rolled
PROBABILITY ASSIGNMENT:
Probability Assignment: a function that assigns to each event E a number between 0 and 1,
which represents the probability of the event E
 Notation: Pr(E)
0 = probability assignment of an IMPOSSIBLE event (empty set)
1 = probability assignment of the CERTAIN EVEN (entire sample space )
Probability Assignment Requirements:
2. The sum of the probabilities of the SIMPLE EVENTS equals 1
EXAMPLE: There are six players in a tennis tournament: Ana (Russian, Female), Ivan
(Croatian, male), Lleyton (Aussie, male), Roger (Swiss, male), Serena (American, female),
and Venus (American, female).
Sample space is who will win the tournament:
Professional Odds-Maker comes up with the following probability assignment:
Pr(Ana) = 0.08 , Pr(Ivan) = 0.16, Pr(Lleyton) = 0.20, Pr(Roger) = 0.25, Pr(Venus) = 0.16
What is the probability of Serena winning?
ALL probabilities are simple events of the addition of simple events.
1) Probability an American will win:
2) Probability a male will win:
3) Probability an American male will win:
4) Probability a European wins:
5) Probability a female will win:
PROBABILITY SPACE: the combination of sample space and its probability
assignment
 Sample Space: S = {σ1, σ2, …, σN }
o σ1, σ2, …, σN represents the simple events of the space
 Probability Assignment: Pr(σ1), Pr(σ2), …, Pr(σN)
o Each of these numbers is between 0 and 1.
o Pr(σ1) + Pr(σ2) + …+ Pr(σN) = 1
 Events: These are all the subsets of S, including {} and S. The probability of
an event is given by the sum of the probabilities of the individual outcomes
that make up the event.
o Pr({}) = 0
o Pr(S) = 1
EXAMPLE Probability Space: Consider the sample space S = {σ1, σ2, σ3, σ4}.
a. If all outcomes are equally likely, then what are their probabilities?
b. If Pr(σ1) = 0.3, Pr(σ2) = 0.25, and Pr(σ3) = .17, then what is Pr(σ4)?
c. If Pr(σ1) = 0.18, Pr(σ2) = 0.12, and Pr(σ3) = Pr(σ4), then what is Pr(σ4)?
d. If Pr(σ1) = 0.18, Pr(σ2) = 0.22, and Pr(σ3) is double Pr(σ4), then what is Pr(σ4)?
e. If Pr(σ1) is double Pr(σ2) and Pr(σ2) = Pr(σ3) = Pr(σ4), then what are all probabilities?
15.4 HOMEWORK – Use a separate sheet
15. 4 EVENTS: p. 534 #42, 44 Write out all events described below in set notation .
42. Consider random experiment where a student takes a four-question true(T)-false(F)
quiz.
a. E1: “exactly 2 of the answers are Ts.”
c. E3: “at most 2 of the answers are Ts.”
b. E2: “at least 2 of the answers are Ts.”
d. E4: “first 2 answers are Ts”
44. Consider random experiment of drawing 1 card out of an ordinary deck of 52 cards.
a. E1: “card drawn is the queen of hearts”
c. E3: “card drawn is a heart”
b. E2: “card drawn is a queen”
d. E4: “card drawn is a face card”
15.4 PROBABILITY ASSIGNMENT: p. 533 #35 – 39
35. Consider the sample space S = {σ1, σ2, σ3, σ4, σ5}. You are given Pr(σ1) = 0.22 and Pr(σ2)
= 0.24.
a. If σ3, σ4, and σ5 all have the same probability, find Pr(σ3).
b. If σ3 has the same probability as σ4 and σ5 combined, find Pr(σ3).
c. If σ3 has the same probability as σ4 and σ5 combined, and if Pr(σ5) = 0.1, give the
probability assignment for this probability space.
36. Consider the sample space S = {σ1, σ2, σ3, σ4}. Suppose you are given Pr(σ1) + Pr(σ2) =
Pr(σ3) + Pr(σ4)
a. If Pr(σ1) = 0.15, find Pr(σ2).
b. If Pr(σ1) = 0.15 and Pr(σ3) = 0.22, give the probability assignment for this
probability space.
37. Seven players are entered in a tennis tournament. According to an expert
handicapper, P2, P3, …, P7 have the same probability of winning, and p1 is twice as likely
to win as one of the other players. Write down the sample space, and find the probability
assignment for the probability space defined by this handicapper’s opinion.
38. Six players are entrees in a chess tournament. According to an expert handicapper, P1
has a probability of 0.25 of winning, P2 has a probability of 0.15 of winning, P3 has a
probability of 0.09 of winning, and P4, P5, and P6 all have an equal probability of winning.
Write down the sample space, and find the probability assignment for the probability
space defined by this handicapper’s opinion.
39. A circular spinner has a RED sector with a central angle of 1080, BLUE and WHITE
sectors both with central angles of 720, and GREEN and YELLOW sectors both with
central angles of 540. Assume that the needle is spun so that it randomly stops at one of
these sectors. Describe the sample space for this game and Give the probability
assignment for this probability space.
SECTION 15.5 EQUIPROBABLE SPACES PART 1
EQUIPROBABLE SPACES: A probability space where each simple event has an equal probability of
occurring (All outcomes are equally likely)
Probability of EVENT, E: The probability of an event E occurring is ratio of the size of event to the size
of the sample space.

Pr( E ) 
Example #1: 12 boys and 15 girls in class. Probability a boy is called out of class is
Example #2: A die being tossed once.
a. What is the sample space, S?
b. What is N(S)?
c. What is the probability of rolling an even number?
d. What is the probability of rolling a 1?
TERMINOLOGY DECK OF CARDS: 52 total Cards
 2 Colors = Red or Black
 4 Suits = Hearts, Diamonds, Spades, Clubs


13 Values = A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K
Face Cards = J, Q, K
COMPLEMENT: a negation of the original statement = NOT EVENT
1) Event = Pick a Black Card
3) Event = Flip a Coin twice and get one tail
2) Event = Pick a Heart
4) Event = Flip 3 times and get all heads
Probability of COMPLEMENT of Event, Ec:

Example 1: 12 boys and 15 girls in class. Probability a boy is not called out of class is

Example 2: The eye color of students is recorded as 150 blue, 285 brown, 93 green, and 72 hazel.
What is the probability that a student doesn’t have blue eyes?

Example 3: If a number from 0 to 999 is to be called out loud. What is the probability that the
number will contain at least one 3?
E and EC are called COMPLEMENTARY EVENTS .
 The probabilities of complementary events add up to 1: Pr(E) + Pr(EC) = 1
 Reminder: Sometimes it’s easier to calculate Pr(EC) instead of Pr(E).
USE COUNTING THEORY TO FIND PROBABILITIES
Example #1: Consider random experiment of tossing a fair coin four times.
a. What is N(S) = total number of possible coin flip combinations?
b. What is the probability of getting EXACTLY one head?
c. What is the probability of getting AT LEAST one head?
Example #2: Bit strings of length 6 chosen at random.
a. How many total bit strings of length 6 are possible?
b. Probability bit string has EXACTLY 3 1’s.
c. Probability bit string has AT MOST 2 1’s.
d. Probability bit string has AT LEAST 3 1’s.
Example #3: A couple is planning to have 3 children and is concerned about their gender.
a. How many different ways could they have boys or girls?
b. What is the probability they will have one boy?
Example #3 CONTINUED…
c. What is the probability they will have two girls?
d. What is the probability they will have at most one boy?
e. What is the probability they will have at least one girl?
Example #4: Draw 2 card from a standard deck of 52, putting the card back in the deck
before redrawing. (REPLACEMENT IS ALLOWED)
a. How many different ways can two cards be drawn?
b. What is the probability to draw 2 kings?
c. What is the probability to draw 2 face cards?
d. What is the probability to draw a red then a black card?
e. What is the probability to draw a red and a black card?
f. What is the probability to draw a two diamonds card?
g. What is the probability to draw a two of same suit?
SECTION 15.5 EQUIPROBABLE SPACES PART 2
INDEPENDENT EVENTS: two events are said to be independent if the occurrence of one
event does not affect the probability of the occurrence of the other.
Example of Independent Events:
#1. Tossing a coin twice times
#2. Rolling a die consecutive times
#3. Choosing a card from a deck, replacing it, and drawing again
#4. 3 red, 6 green, and 5 yellow marbles are chosen from a bag and a marble is replaced after it
was drawn.
(Key Concept = REUSING SOMETHING/ REPLACEMENT)
The Multiplication Principle for Independent Events
If events E and F are independent, then the probability that both occur (E and then F) is the
product of the respective probabilities.

Pr( E and then F )
Comparison of Independent and Dependent Multiplication:
DEPENDENT
A deck of cards is shuffled and two cards
are dealt. What is the chance that both are
aces?
Probability (2 Aces) =
INDEPENDENT
A deck of cards is shuffled, a card is drawn,
and then replaced in the deck, shuffled, and
drawn again. What is the chance that both
are aces?
Probability (2 Aces) =
Example #1: 7 red marbles, 5 green marbles, and 8 yellow marbles are in a bag and each
time a marble is chosen it is replaced back in the bag for the next draw.
a. Find Pr(Red then Yellow)
d. Find Pr(Red then Red)
b. Find Pr(Yellow then Red)
e. Find Pr(Red then Green)
c. Find Pr(Red and Yellow)
f. Find Pr(Green then Yellow)
Example #3: 7 red marbles, 5 green marbles, and 8 yellow marbles are in a bag
and drawn at random with no replacement.
a. Find Pr(Red then Yellow)
c. Find Pr(Red and Yellow)
b. Find Pr(Yellow then Red)
d. Find Pr(Red then Red)
Example #4 : Roll a die 4 times. What is the probability of getting exactly 3 1’s
to show up?
BINOMIAL FORMULA:
Probability that an independent event will occur exactly k times out of n tries =
C k p (1  p )
k
n
nk




n = number of total trials (tosses of coin, roll of dice, etc)
k = EXACT number of times your outcome will occur
p = probability event will occur at once (single event)
1 – p = probability event will not occur at one trial (COMPLEMENT)

n
C k = choosing k of the n trials to be the event you want
(ways it can occur in all)
Ex #5a: Roll a die 6 times. What is the probability of getting exactly 3 1’s to show up?
Ex #5b: Toss a coin 4 times. What is the probability of getting exactly 3 H’s to
show up?
Ex #6: Draws are made at random with replacement from the box containing 8
identical balls marked with {1, 1, 2, 3, 3, 3, 4, 5}.
a. Probability of exactly 20 1’s after 25 draws.
b. Probability of exactly 8 3’s after 16 draws.
c. Probability of exactly 5 5’s after 15 draws.
Equiprobable Spaces Homework: pp. 534 – 535 #47 – 56
47) Consider the random experiment of tossing an honest coin 3 times in a row. Find the probability.
a. E1: “Tossing exactly 2 heads”
c. E3: “half of the tosses are heads and half are tails”
b. E2: “all tosses come out the same”
d. E4: “first two tosses are tails”
48) Consider the random experiment where a student takes a 4-question true or false quiz. Assume now that
the student randomly guesses the answer for each question. Find the probability.
a. E1: “exactly 2 of the answers given are Ts”
c. E3: “at most 2 of the answers given are Ts”
b. E2: “at least 2 of the answers given are Ts”
d. E4: “first 2 answers given are Ts”
49) Consider the random experiment of rolling a pair of honest dice. Let T2, T3, …, T12 represent the events
of rolling “a total of 2”, “a total of 3”, .., “a total of 12”. (Hint: See Exercise 43)
a. Find Pr(T6) and Pr(T8)
d. E2: Roll a total of 3 or less; Find Pr(E2)
b. Find Pr(T5) and Pr(T9)
e. E3: Roll a total of 7 or 11; Find Pr(E3)
c. E1: Roll two of a kind; Find Pr(E1)
50) Consider the random experiment of drawing 1 card out of an honest deck of 52 cards. Find the
probability.
a. E1: “the card drawn is the queen of hearts”
c. E3: “the card drawn is a heart”
b. E2: “the card drawn is a queen”
d. E4: “the card drawn is a face card”
51) Consider the random experiment of tossing an honest coin 10 times in a row. Find the probability.
a. E1: “toss no tails”
c. E3: “toss exactly twice as many heads as tails”
b. E2: “toss exactly 1 tail”
52) Consider the random experiment of drawing two cards out of an honest deck of 52 cards. Find the
probability. The order of the cards does not matter.
a. E1: “draw a pair of queen”
b. E2: “draw a pair”
53) If a pair of honest dice are rolled once, find the probability of
a. rolling a total of 8
c. rolling a total of 8 or 9
b. not rolling a total of 8
d. rolling a total of 8 or more
54) A gumball machine has gumballs of four different flavors: A, B, C, and D. When a 50-cent piece is put
into the machine, five random gumballs come out. Find the probability that
a. each gumball is a different flavor
b. at least two of the gumballs are the same flavor.
55) A student takes a 10-question true or false quiz and randomly guesses the answer to each question.
Suppose a correct answer is worth 1 point, an incorrect answer is worth -0.5 points. Find the probability that
the student
a. gets 10 points.
c. gets 8.5 points.
e. gets 5 points.
b. gets -5 points.
d. gets 8 or more points.
f. gets 7 or more points.
56) Ten names (A, B, C, D, E, F, G, H, I, J) are written on separate slips of paper, put in a hat, and mixed
well. Four names are randomly taken out of the hat, one at a time. Assume that the order in which the names
are drawn matters. Find the probability that
a. A is the first name chosen
b. A is one of the four names chosen
c. A is not one of the four names chosen
d. the four names chosen are A, B, C, D in that order
15.6 ODDS
ODDS: For an arbitrary event, odds represent a comparison of the number of ways
than even can occur (favorable outcomes) to the number of ways an event does not
occur (unfavorable outcomes).
 Favorable Outcomes = N(E)
 Unfavorable Outcomes = N(Ec)
The odds of (odds in favor of) the event E are given by the ratio N(E) to N(Ec).
The odds against the event E are given by the ratio N(Ec) to N(E).
Example #1 - Odds: Suppose you are playing a game in which you roll a pair of honest dice.
If you roll a total of 7 or 11, you win the game. What are your odds of winning in this game?
What are your odds of losing?
Win: Odds in Favor of
Lose: Odds against
Example #2– Connecting Probability and Odds:
#2a: Steve Nash shoots free throws with a probability of 0.90. For every 100 free throws,
Nash will make 90 and will miss 10.
SIMPLIFY SIZES:
(GCF)
The odds of Nash making a free throw are
The odds against Nash making a free throw
#2b: Shaquille O’Neal shoots free throws with a probability of 0.52. For every 100 free
throws, Shaq will make 52 and will miss 48.
SIMPLIFY SIZES:
(GCF)
The odds of Shaq making a free throw are
The odds against Shaq making a free throw are
Example #3: There are six players in a tennis tournament: Ana, Ivan, Lleyton, Roger,
Serena, and Venus.
Express each of these probabilities as odds of winning the tourney and odds against winning
the tourney.
Probability (Find a reduced fraction for the probability)
Odds For
Odds Against
Pr(Ana) = 0.08
Pr(Ivan) = 0.16
Pr(Lleyton) = 0.20
Pr(Roger) = 0.25
Pr(Serena) = 0.15
Pr(Serena) = 0.16
Example #4: Find the probability of event E for each of the given odds.
a. The odds of event E are 7 to 8
d. The odds against event E are 7 to 3.
b. The odds of event E are 2 to 15
e. The odds of event E are 9 to 11.
c. The odds against event E are 13 to 7.
f. The odds against event E are 1 to 1.
g. The odds of E are the same as the odds against E.