The Basics of Probability Theory
MATH 102
Contemporary Math
S. Rook
Overview
• Section 14.1 in the textbook:
– Sample spaces & events
– Counting & probability
– Odds
Sample Spaces & Events
Sample Spaces
• Random phenomena: an occurrence that
varies between several different outcomes
– e.g. Weather, rolling dice, flipping coins etc.
• An experiment is the observation of a random
phenomena and noting the outcomes
• Sample space: the set of ALL possible
outcomes for an experiment
– e.g. List the sample space if we spin a spinner
with three regions: red, white, and blue
Probability & Events
• Probability is a measure of the likelihood of an
outcome in the sample space occurring
– The probability for an outcome is always between 0
and 1 inclusive
• An outcome with a probability of 0 will NEVER occur
• An outcome with a probability of 1 will ALWAYS occur
• An event is a subset of the possible outcomes in the
sample space
– The probability of an event, denoted P(E), is the sum
of the probabilities of the individual outcomes that
make up the event
Sample Spaces & Events (Example)
Ex 1: A spinner has three regions: red, white,
and blue. The spinner is spun twice and the
results recorded.
a) List the sample space
b) List those where red appears exactly once.
c) List those outcomes where blue appears at least
once.
d) List those outcomes where yellow does not
appear at all
Sample Spaces & Events (Example)
Ex 2: Suppose that we draw a card from a
standard 52-card deck which is our sample
space. List the event where we draw:
a) A 6 from the deck.
b) A red face card from the deck.
Counting & Probability
Counting & Probability
• Given that outcomes in the sample space are
equally likely, the probability of event E is
a
nE  where n(E) and n(S) represent the
P E  
ardinalnS  number of elements in the event
set and sample space set respectively
– For this class, each outcome in a sample space will
be equally likely
– Possible that we may need the F.C.P.,
permutations, or combinations to find n(E), n(S),
or both
Counting & Probability (Example)
Ex 3: We are simultaneously rolling two four-sided
dice having the numbers 1, 2, 3, and 4 on their faces.
Outcomes in the sample space are listed as pairs
such as (1, 2).
a) How many elements are in the sample space?
b) What is the probability of the sum of the rolls
being even?
c) What is the probability that the sum of the rolls is
greater than six?
Counting & Probability (Example)
Ex 4: Use the following table which relates living
arrangements and GPA for 320 students:
On Campus
At Home
Apartment
Totals
Below 2.5
98
40
44
182
2.5 to 3.5
64
25
20
109
Over 3.5
17
4
8
29
Totals
179
69
72
320
a) If we select a random student, what is the probability
that the student has a GPA of at least 2.5?
b) If we select a random student, what is the probability
that the student lives off-campus?
Counting & Probability (Example)
Ex 5: In a given year, 2,048,861 males and
1,951,379 females were born in the U.S. If a
child is selected randomly from this group,
what is the probability that the child is a
female?
Odds
Odds
• Odds is another commonly used concept that utilizes
probability
– Odds are often associated with betting
• e.g. If you win a bet having 6 : 1 odds, you will receive
$6 for every $1 that you bet
• The odds against an event happening is the ratio of the
probability of the event not happening and the
probability of the event happening
– Normally expressed in colon notation (a : b)
– e.g. Consider the odds 6 : 1 again
• 6 out of 7 times event E would not happen
Odds (Continued)
• Consider a sample space with 10 outcomes
– If 6 outcomes satisfy an event E, what is P(E)?
– How many outcomes would NOT satisfy E? What is
this probability?
– What are the odds against E?
Odds (Example)
Ex 6: If we draw one card from a standard 52card deck:
a) What is the probability of drawing a spade?
b) What are the odds against drawing a
spade?
Odds (Example)
Ex 7: Solve:
a) If the odds against event E are 5 : 2, what is
the probability of E?
b) What are the odds against the Yankees
winning the World Series if the probability of
them winning the world series is 0.30?
Summary
• After studying these slides, you should know
how to do the following:
– List and/or count the elements of the sample
space of an experiment
– Calculate the probability P(E) of event E occurring
– Calculate the odds of event E not occurring
• Additional Practice:
– See problems in Section 14.1
• Next Lesson:
– Complements & Unions of Events (Section 14.2)