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Unit 7A
Fundamentals of Probability
BASIC TERMINOLOGY IN
PROBABILITY
• Outcomes are the most basic possible results of
observations or experiments.
• An event consists of one or more outcomes that
share a property of interest.
• The probability of an event, expressed as
P(event), is always a number between 0 and 1
(inclusive).
• Many times upper case letters are used to
represent events. For example, we could say that
A is the event of getting “heads” when a coin is
tossed. So, P(A) would be the probability of
getting “heads” when a coin is tossed.
PROBABILITY LIMITS
• The probability of an
impossible event is 0.
• The probability of an even that
is certain to occur is 1.
• Any event A has a probability
between 0 and 1, inclusive.
THE MULTIPLICATION
PRINCIPLE
For a sequence of two events in which the first
event can occur M ways and the second event can
occur N ways, the events together can occur a
total of M × N ways.
This generalizes to more than two events.
EXAMPLES
1. How many two letter “words” can be formed if the
first letter is one of the vowels a, e, i, o, u and the
second letter is a consonant?
2. OVER FIFTY TYPES OF PIZZA! says the sign as
you drive up. Inside you discover only the choices
“onions, peppers, mushrooms, sausage, anchovies,
and meatballs.” Did the advertisement lie?
3. Janet has five different books that she wishes to
arrange on her desk. How many different
arrangements are possible?
4. Suppose Janet only wants to arrange three of her
five books on her desk. How many ways can she
do that?
THREE TYPES OF
PROBABILITIES
There are three different types of probabilities.
• Theoretical Probabilities
• Empirical Probabilities
• Subjective Probabilities
THEORETICAL PROBABILITY
A theoretical probability is based on a model in
which all outcomes are equally likely.
It is determined by dividing the number of ways
an event can occur by the total number of possible
outcomes. That is,
number of ways A can occur
P( A) 
total number of outcomes
EXAMPLE
Find the probability of rolling a “7” when a pair
of fair dice are tossed.
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EMPIRICAL PROBABILITY
An empirical probability is based on
observations or experiments. It is the relative
frequency of the event of interest.
In baseball, batting averages are empirical
probabilities.
COMPUTING AN EMPIRICAL
PROBABILITY
Conduct (or observe) a procedure a large number
of times, and count the number of times that
event A actually occurs. Based on these actual
results P(A) is estimated as follows:
number of times A occurred
P( A) 
number of times trial was repeated
EXAMPLE
A fair die was tossed 563 times. The number
“4” occurred 96 times. If you toss a fair die,
what do you estimate is the probability is for
tossing a “4”?
SUBJECTIVE PROBABILITY
A subjective probability is an estimate based on
experience or intuition.
EXAMPLE: An economist was asked “What is
the probability that the economy will fall into
recession next year?” The economist said the
probability was about 15%.
PROBABILITY OF AN EVENT
NOT OCCURRING
Suppose the probability of an event A is P(A).
Then the probability that the event A does not
occur is 1 − P(A).
PROBABILITY DISTRIBUTION
A probability distribution represents the
probabilities of all possible events. It is
sometimes put in the form of a table in which
one column lists each event and the other
column lists each probability. The sum of all
the probabilities must be 1.
MAKING A PROBABILITY
DISTRIBUTION
To make a probability distribution:
Step 1: List all possible outcomes.
Step 2: Identify the outcomes that represent
the same event. Find the probability of each
event using the theoretical method.
Step 3: Make a table in which the first
column lists each event and the second
column lists the probability of the event.
EXAMPLES
1. Suppose you toss a coin three times. Let x be
the total number of heads. Make a table for the
probability distribution of x.
2. Suppose you throw a pair of dice. Let x be the
sum of the numbers on the dice. Make a table
for the probability distribution of x.
EVENT
2
3
4
5
6
7
8
9
10
11
12
PROB.
1/36 ≈ 0.028
2/36 ≈ 0.056
3/36 ≈ 0.083
4/36 ≈ 0.111
5/36 ≈ 0.139
6/36 ≈ 0.167
5/36 ≈ 0.139
4/36 ≈ 0.111
3/36 ≈ 0.083
2/36 ≈ 0.056
1/36 ≈ 0.028
Total
36/36 = 1.000
ODDS
The odds for (or odds in favor of) an event A
are
P( A)
odds for event A 
P(not A)
The odds against (or odds on) an event A are
P(not A)
odds against event A 
P( A)
ODDS IN GAMBLING
In gambling, the “odds on” usually expresses
how much you can gain with a win for each
dollar you bet.
EXAMPLE: At a horse race the odds on the
horse Median are given as 5 to 2. If you bet $12
on Median and he wins, how much will you
gain?