SECTION 15.4 PROBABILITY SPACES
EVENT: any subset of the sample space or any set of individual outcomes
 Simple Event:

Impossible Event:

Certain Event:
EXAMPLE #1: Consider the random experiment of tossing a coin three times.
SAMPLE SPACE:
{HHH, THH, HTH, HHT, TTH, THT, HTT, TTT}
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) = probability of event E
0 = probability assignment of an IMPOSSIBLE event (empty set)
1 = probability assignment of the CERTAIN EVENT (entire sample space )
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?
Probability represents the probabilities of simple events, and all other probabilities follow
by addition.
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 the sample space and its probability assignment
 Sample Space: S = {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: The probability of an event is given by the sum of the probabilities of the
individual outcomes that make up the event.
o Impossible: Pr({}) = 0
o Certain: 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?
HOMEWORK: p. 533 – 534 #35 – 39, 42, 44
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 is RATIO of the size of event to the size of the sample space.
Example #1: 12 boys and 15 girls in class. What is probability a boy is called out of class?
Example #2: A die being tossed once.
a. What is the probability of rolling a 1?
b. What is the probability of rolling an even number?
Example #3: A four character password is made from numbers and letters which are not case sensitive.
a. What is the probability of a password being
b. What is the probability of a password begin
all numbers?
with two letters and end with two numbers?
COMPLEMENT: a negation of the original statement
1) Event = Pick a Black Card
4) Event = Flip a Coin twice and get one tail
2) Event = Pick a Heart
5) Event = Flip a Coin 3 times and get all heads
3) Event = 4 question true or false test with at most 6) Event = Roll a die 6 times and get at least 4 even
2 true responses
digits.
E and EC are called COMPLEMENTARY EVENTS .
 The probabilities of complementary events add up to 1: Pr(E) + Pr(EC) = 1
size of event E
C
c
Pr(
E
)

1

Pr(
E
)

1

Probability of COMPLEMENT of Event, E :
size of sample space S

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?
USE COUNTING THEORY TO FIND PROBABILITIES
#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 rolling getting EXACTLY one head?
c. What is the probability of getting AT LEAST one head?
#2: 6 question true or false test.
a. How many different possible answer keys exist?
b. Probability an answer key has EXACTLY THREE Trues.
c. Probability an answer key has AT MOST TWO Falses.
d. Probability an answer key has AT LEAST THREE Falses.
#3: A four character password is made from numbers and letters which is case sensitive.
a. How many possible passwords are possible?
b. Probability that a password will use no even digits
c. Probability that a password will start with a number and end with a number.
d. Probability that a password will only lowercase letter
#4: Draw two cards from a standard deck, 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?
#5: Draw two cards from a standard deck, holding onto a card after its drawn.
a. How many different ways can two cards be drawn?
b. What is the probability to draw two 5’s?
c. What is the probability to draw two cards with number values?
d. What is the probability to draw two black cards?
e. What is the probability to draw two cards 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 and replaced after each draw.
(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 )  Pr( E )  Pr( 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)
Find Pr(Red then Red)
b. Find Pr(Yellow then Red)
d. Find Pr(Red then Green)
c. Find Pr(Red and Yellow)
e. Find Pr(Green then Yellow)
Example #2: 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 #3 : Roll a die 4 times. What is the probability of getting exactly 3 1’s
to show up?
BINOMIAL PROBABILITY 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.
Homework: pp. 534 – 535 #47 – 56
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. How many ways can you lose and win the
game?
What are your odds of winning in this game?
Win: Odds in Favor of
What are your odds of losing?
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
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
The odds against Shaq making a free throw
Example #3: 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.