Chapter 8:
Introduction to Probability
• Probability measures the likelihood, or the
chance, or the degree of certainty that some
event will happen.
• The study of probability provides us with tools for
measuring and analyzing uncertainties associated
with the occurrence of future events. It allows us
to make numerical statements about how likely or
unlikely some future event is to occur.
• Probability ranges from 0 to 1, with events closer
to 0 being very unlikely to occur, and events closer
to 1 being almost likely to occur.
Terminology
• An experiment is any phenomenon whose
occurrence yields an observable but
unpredictable result.
– Examples: tossing a coin and recording what face
turned up
throwing a six sided die and observing what
number was rolled
• Sample space refers to the set of all possible
outcomes of an experiment.
– Examples: tossing a coin: S= {head, tail}
rolling a die: S= {1,2,3,4,5,6}
Law of Large Numbers
• The law of large numbers states that if we
observe more and more repetitions of any chance
process, the proportion of times that a specific
outcome occurs approaches a single value.
• Think about flipping a coin. You might flip a coin
5 times and get a tail 4 out of those 5 times. That
doesn’t mean the probability of getting a tail
when you flip a coin is 0.50. But as your sample
size increases, the probability will approach 0.50.
Classical (Theoretical) Probability
• In theoretical probability, an assumption is made
that the outcomes are equally likely and mutually
exclusive, meaning that no two events can occur
together (for example flipping both a head and a
tail on the same flip).
• The probability of event A occurring is defined by
the formula
Example: A six-sided die is rolled once. What is the probability that
a 5 is rolled?
Empirical Probability
• In empirical probability, outcomes are not
equally likely, and thus their weight needs to
be accounted for.
• The calculation of empirical probability is
based on the assumption that the proportion
of times an event actually occurred in the past
reflects a blueprint that will be operative in
the future.
Example: Dale has a closet with 4 white shirts, 6
blue shirts, and 5 gray shirts. If he randomly
selects a shirt from a dark closet, what is the
probability that the shirt is white?
Example: Brennan has a jar filled with skittles.
There are 10 red skittles, 6 purple skittles, and 8
green skittles. If he randomly reaches into the
jar, what is the probability that he draws a green
skittle?
Mutually Exclusive Events
Example: Find the probability of randomly selecting
the following student(s).
a) P(junior)
b) P(sophomore and senior)
c) P(sophomore or senior)
Example: A six-sided die is rolled once. Find the
probability of each event occurring.
a) P(6)
b) P(4 or 5)
c) P(even number)
d) P(2 and 6)
Non-Mutually Exclusive Events
• Two events are non-mutually exclusive if they
could (but do not have to) occur
simultaneously.
• The common area where A and B occur is
called their intersection.
The General Addition Rule
• The notation “A or B” denotes the event that
“either A or B or both occur.”
• To find P(A or B) we add P(A)+P(B) and then
subtract the probability that both events
occur at the same time (overlap).
• P(A or B) is referred to as the “union.”
Example: 100 workers were sampled to find if they
favor or oppose a certain company issue. If one
worker is selected at random, find the following
probabilities.
a) P(blue-collar)
0.60
b) P(white-collar)
0.40
0.64
c) P(favored the issue)
Note: This type of table is known as a two-way
table, or a contingency table.
• Are the events “favor” and “blue collar” mutually
exclusive?
• In order for this to occur, there has to be no
common elements, meaning
• However, we know that
• Therefore, the events are not mutually exclusive.
Above is the results of 81 regular season home baseball
games and whether or not fans received a free taco. If a
game is randomly selected, find the probability that:
Complementary Events
• The probability that an event does not occur is 1
minus the probability that the event does occur.
• For example, if event A occurs 70% of the time, it
will fail to occur 30% of the time.
• We refer to event “not A” as the complement of
A.
Example:
Example: Two coins are tossed. What is the
probability that “two heads do not show?”
S={HH, HT, TH, TT}
Independent vs Dependent Events
• Events A and B are independent if the occurrence
of either one of them does not affect the
probability of occurrence of the other.
• The result of the second event does is not
affected by the result of the first event.
• For example, if a coin is tossed twice in
succession, the probability of obtaining a head on
the second toss is not in any way influenced by
the outcome of the first toss.
In 2010, Jose Bautista of the
Toronto Blue Jays led the
Major Leagues with 54 home
runs. The table above
summarizes the outcomes of
the Blue Jays’ 81 home games
and whether or not Bautista
hit at least 1 home run in the
game. Question: Are the
events “Bautista hits a
homerun” and “Blue Jays
win” independent?
The events are not independent.
Clearly the Blue Jays were more
likely to win when Bautista hit a
home run.
Dependent Events
• Two events A and B are dependent on each
other if the occurrence of one does affect the
probability of occurrence of the other.
• As we saw on the previous slide, Bautista
hitting a home run and the Blue Jays winning
the game were dependent events, as the Blue
Jays were clearly more likely to win the game
if Bautista homered as opposed to if he did
not.
• Conditional probability describes the
probability that an event occurs, given that we
know that a different event has already
occurred.
• Let’s revisit our taco example. If we randomly
select one of the victories, what is the probability
that a fan at that game received a free taco?
To find this probability, we only care about games in which the team
won. A win must occur first. Given the win occurred, what is the
probability the fans got a free taco. It may help to eliminate the
loss column since we don’t care about games in which the team
lost.
• You try this one. Find the probability that the
team won the game, given that free tacos
were given away.
Expected Value
• A random variable takes on numerical values
that describe the outcomes of a chance
process.
• In general, for a random variable X, the mean
value of X (also called the expected value of X)
can be found by multiplying each value of X by
its probability and then adding together the
products.
Example: The table above uses the results of the
World Series from 1945 to 2010 to estimate the
probability distribution of X. This probability
distribution lists the possible number of games and
how often those values occurred. On average, how
many games does a World Series last? In other
words, what is the mean of the random variable X?
Ex. 2: Hole #13 at the Augusta National golf course
is one of the most famous holes in golf. Lined with
the course’s signature azaleas (not Iggy), this hole is
also a favorite of players for its relative ease. The
hole is a par 5, meaning that professional golfers
would be expected to complete the hole in 5
strokes. Let X = the score on hole #13 for a
randomly selected golfer on day 1 of the 2011
Masters. Above is the probability distribution of X.
Calculate the expected value of X.