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Lecture 3.3
Contemporary Mathematics
Instruction: Theoretical & Empirical Probability
Probability is a numerical measure of the likelihood of an outcome or event associated
with an experiment. Specifically, probability is a ratio. To address probability, this lecture will
consider two experiments. First, the flip of a coin that can land head-side up or tail-side up.
Second, the birth of a child that can be born male or female.
The coin flip experiment has two possible outcomes: the coin lands head-side up,
denoted h, and the coin lands tail-side up, denoted t. The sample space, S, for the coin flip is
S = {h, t} . Let H represent the event that the coin lands head-side up, then H = {h} . Let T
represent the event that the coin lands tail-side up, then T = {t} . A typical question associated
with this experiment asks, "How likely is event H?" To answer this question, we assign a
measure called the probability of H, denoted P ( H ) , based on the definition of probability as the
ratio of the number of favorable outcomes to the number of total outcomes, i.e.,
P ( H ) = n ( H ) n ( S ) , so P ( H ) = 1 2 . This definition of probability requires that all the
outcomes be equally likely. Applied to the coin flip experiment, the likelihood of each possible
outcome intuitively seems equal and no actual flip of the coin has to be performed. This type of
probability is called theoretical probability since the likelihoods are based on theory rather than
actual performed experiments. Formally, Theoretical Probability is defined below.
For a finite sample space S of equally likely outcomes, the theoretical
probability of event E, a subset of S, is given by
P(E) =
number of elements (outcomes) in E n ( E )
.
=
number of elements (outcomes) in S n ( S )
Not all probabilities are theoretical. Some are based on the performance of experiments.
Consider, for example, our second experiment, the birth of a child that can be born male or
female. According to Mark Alpert writing in Scientific American in July 1998, there were 74
children born in the years 1977 to 1984 to parents who were exposed to a cloud of dioxin
released into the atmosphere after a chemical plant explosion in Seveso, Italy in 1976. Of these
74 births, 26 of the infants were boys. The other 48 infants were girls. These 74 births reported
by Mark Alpert represent 74 trials of an experiment. The set of actual outcomes of each of the
trials is called a trial space.
A trial space, T , is a set of outcomes recorded from a finite set of trials.
In this example, T = {b1 , b2 , … , b26 , g1 , g 2 , … , g 48 } , and n ( T
) = 74.
The trials that resulted in
boy births are a special subset of the trial space called an event space. For example, if B
represents the event space that includes all the outcomes from the finite set of trials in which a
Lecture 3.3
boy was born, B = {b1 , b2 , …, b26 } , and n ( B ) = 26 . Similarly, if G represents the event space
that includes all outcomes from the finite set of trials in which a girl was born, then
G = { g1 , g 2 , …, g 48 } , and n ( G ) = 48 . Based on these births, it is obvious that the likelihood of a
boy being born under the given conditions is not equal to the likelihood of a girl being born. For
such instances, the definition of theoretical probability does not apply. Instead, we will rely on
the definition of Empirical Probability stated in the box on the following page.
For an outcome E that may happen when an experiment is performed, then
the empirical probability of the outcome E is given by
P(E) =
number of trials with outcomes that describe E n ( E )
=
n (T )
number of trials
where E is the event space that contains all the outcomes from a finite set of
trials that describe event E.
The two definitions of probability are similar and the distinction between the two is
subtle, having mostly to do with the difference between a sample space and a trial space.
Theoretical probability depends in part on the definition of a sample space, which is the set of all
the equally likely possible outcomes associated with an experiment. This is a set of outcomes
that could occur for a given experiment. Empirical probability relies in part on the definition of a
trial space, which is the set of all the actual outcomes that have occurred in a finite set of trials of
an experiment.
Let's reconsider the trials comprised of the 74 births in the years 1977 to 1984 to parents
exposed to dioxin due to the chemical plant explosion in Seveso, Italy in 1976. Here, it is
evident that the definition for empirical probability applies because experiments have been
performed and outcomes have occurred. If B represents the event that a boy was born under the
stated conditions of the dioxin-exposure experiment, then we find P(B), the probability of event
B, using the definition of empirical probability:
P ( B) =
number of trials with outcomes that describe B n ( B ) 26 13
=
=
=
≈ 0.351 ≈ 35.1% .
number of trials of experiment
n ( T ) 74 37
The Law of Large Numbers stated below makes a connection between empirical and
theoretical probability.
As an experiment undergoes an increasing number of trials, the empirical
probability of a particular event approaches a fixed number. More specifically, as
n ( T ) approaches infinity, the empirical probability of an event approaches the
theoretical probability of the event.
Lecture 3.3
The Law of Large Numbers allows us to estimate the theoretical probability of an event using
experiments, multiple trials, and empirical probability. The Law of Large Numbers indicates that
the estimate becomes more accurate as the number of trials increases. If, for example, there had
been 10,000 births instead of only 74 under the stated dioxin-exposure conditions, the percent of
boy births would be even closer to the theoretical probability of giving birth to a boy after such
exposures.
Instruction: Properties of Probability
Recall that events are subsets of a sample space. Accordingly, event E is a subset of S
and 0 ≤ n ( E ) ≤ n ( S ) . Using this inequality, we can arrive at a fundamental property of
probability. Consider dividing the terms of the inequality, 0 ≤ n ( E ) ≤ n ( S ) , by n ( S ) :
n(E) n(S )
0
≤
≤
.
n(S ) n(S ) n(S )
Note that zero divided by any non-zero number is zero and any non-zero number divided by
itself is one:
n(E)
0≤
≤ 1.
n(S )
Recall that theoretical probability defines P ( E ) as n ( E ) n ( S ) :
0 ≤ P ( E ) ≤ 1.
Using the same process using the definition of empirical probability and the fact that an event
space is a subset of a trial space, we can arrive at the same conclusion for empirical probability.
Accordingly, probability is a measure from zero to one inclusive.
In summary, we see that the probability of an event must be a number r such that zero is
less than or equal to r and r is less than or equal to one. Moreover, if E is an event that does not
or cannot occur in the experiment or is an impossible event, then P ( E ) = 0 . If, on the other
hand, E is the only event that occurs in the experiment or is an event certain to occur, then
P ( E ) = 1 . The box below summarizes these properties.
If E is an event associated with an experiment, then
I.
0 ≤ P ( E ) ≤ 1.
II.
If E = {
III.
} or if E = { } , then P ( E ) = 0 .
If E = S or if E = T , then P ( E ) = 1 .
Application Exercise 3.3
Problems
Human eye color is controlled by a single pair of genes (one paternal gene coupled with one
maternal gene). This pair of genes is called a genotype. Brown eye color, B, is dominant over
blue eye color b. Therefore, in the genotypes Bb the brown gene dominates and the person
inheriting the Bb genotype has brown eyes.
#1
Assume that two parents carry a Bb genotype and that their offspring is equally likely to
inherit the B gene as they are the b gene. What is the theoretical probability that a child
of these parents will have blue eyes?
#2
Assume that two parents carry a Bb genotype and have had eight children, seven with
brown eyes and one with blue eyes. What is the empirical probability that one of their
children has blue eyes?
#3
Assume that one parent carries a bb genotype and that the other parent carries a Bb
genotype. What is the theoretical probability that a child of these parents will have blue
eyes?
#4
Due to a trait called co-dominance, some children that inherit a Bb genotype have eyes
that are neither brown or blue but hazel. Assume that past experimentation has shown
that three of every one-hundred babies with the Bb genotype have hazel eyes, what is
the empirical probability that a child with the Bb genotype will have hazel eyes?
#5
Consider a sample space with three elements: a1 , a2 , and a3 . If P ( a1 ) = 0.3 and
P ( a2 ) = 0.3 , find P ( a3 ) .
#1 Let L represent the event that child has blue eyes, then P ( L ) =
#2 P =
1
= 0.125
8
#3 Let L represent the event that child has blue eyes, then P ( L ) =
3
= 0.03
100
#5 P ( a3 ) = 0.4
#4 P =
1
= 0.25 .
4
1
= 0.5 .
2
Assignment 3.3
Problems
#1
Suppose a jar contains twenty-five marbles: five red, ten blue, and ten white. Let W
represent the event that a white marble is selected when one marble is selected at
random. Find P (W ) .
#2
A fair, six-sided die with faces numbered one through six is rolled one-hundred times
and a six comes up twenty times. In this experiment, what is the empirical probability
of rolling a six? What is the theoretical probability of rolling a six with the die? Why
are the two answers different?
#3
If a coin is tossed and then a single six-sided die is rolled, list the sample space for this
experiment.
#4
Consider the sample space above from the experiment where a coin is tossed and then a
single six-sided die is rolled. Let T ∩ E represent the event that the coin lands tail-side
up and the die lands with an even number facing up. Find P (T ∩ E ) .
#5
Consider a sample space with four elements: a1 , a2 , a3 , and a4 . If P ( a1 ) = 0.3 ,
P ( a2 ) = 0.2 , and P ( a3 ) = 0.4 , find P ( a4 ) .
Lecture 3.4
Contemporary Mathematics
Instruction: Probability Distribution
This lecture discusses probability distributions. In tabular form, a probability distribution
is a table of two rows (or columns). One row lists the mutually exclusive events whose union
comprises the sample space of an experiment. The second row lists the probabilities associated
with each event of the experiment. Consider an experiment that entails rolling the twelve-sided
die shown in Figure 1 and observing the number that lands face up.
Figure 1
The sample space for the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} . To construct the
probability distribution, we assign a random variable, X, to represent events under observation.
One row will list the "values" of X and the other will list the corresponding probabilities, P ( X ) .
X
P(X )
1
1 12
2
1 12
3
1 12
4
1 12
5
1 12
6
1 12
7
1 12
8
1 12
9
1 12
10
1 12
11
1 12
12
1 12
Since X represents the mutually exclusive events whose union comprises the sample space, the
sum of the probabilities equals one.
The experiment above was comprised of one toss. Consider now an experiment
comprised of two tosses: rolling two six-sided die like those in Figure 2, observing the sum of
the result of each die.
Figure 2
Let A be the set of outcomes associated with a single toss of a six-sided die, A = {1, 2, 3, 4, 5, 6} ,
then the sample space of the experiment equals the Cartesian product, A × A .
⎧(1,1) , (1, 2 ) , (1, 3) , (1, 4 ) , (1, 5 ) , (1, 6 ) , ⎫
⎪
⎪
⎪( 2,1) , ( 2, 2 ) , ( 2, 3) , ( 2, 4 ) , ( 2, 5 ) , ( 2, 6 ) , ⎪
⎪
⎪
⎪( 3,1) , ( 3, 2 ) , ( 3, 3) , ( 3, 4 ) , ( 3, 5 ) , ( 3, 6 ) , ⎪
S = A× A = ⎨
⎬
⎪( 4,1) , ( 4, 2 ) , ( 4, 3) , ( 4, 4 ) , ( 4, 5 ) , ( 4, 6 ) , ⎪
⎪ 5,1 , 5, 2 , 5, 3 , 5, 4 , 5, 5 , 5, 6 , ⎪
⎪( ) ( ) ( ) ( ) ( ) ( ) ⎪
⎪( 6,1) , ( 6, 2 ) , ( 6, 3) , ( 6, 4 ) , ( 6, 5 ) , ( 6, 6 ) ⎪
⎩
⎭
Lecture 3.4
The defined experiment observes the sum of the result of each die, so the random variable X will
represent a sum of the ordered pairs.
X
P(X )
2
1
36
3
2
1
=
36 18
3
1
=
36 12
4 1
=
36 9
5
36
6 1
=
36 6
5
36
4 1
=
36 9
3
1
=
36 12
2
1
=
36 18
1
36
4
5
6
7
8
9
10
11
12
Graphically, a probability distribution is a set of ordered pairs ( X , P ( X ) ) .
P(X )
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
2
3
4
5
6
7
X
8
9
10
11
12
Application Exercise 3.4
Problems
#1
Consider an experiment that involves counting the number of defective light bulbs in a
case of twenty-four. Use either roster or set-builder notation to name the set of possible
values for the random variable associated with this experiment.
#2
Suppose NASA has developed a test to measure microgravity tolerance in its
astronauts. The test has possible scores of 0, 1, 2, 3, 4, and 5. Suppose the test is
administered to twenty-five astronauts with eight astronauts scoring 5, ten astronauts
scoring 4, three astronauts scoring 3, two astronauts scoring 2, one astronaut scoring 1,
and one astronaut scoring 0. Now consider an experiment where a astronaut from this
group is chosen at random and the astronaut's microgravity tolerance score is observed.
Construct a probability distribution for this experiment.
For the remaining problems, suppose NASA has developed a test to measure isolation tolerance
in its astronauts and that the test has possible scores of 1, 2, 3, 4, 5, 6, and 7. Suppose further
that NASA creates seven-member crews such that every possible isolation tolerance score is
represented in the crew (i.e., one member has a score of 1, another has a score of 2, etc. up to 7).
#3
Consider an experiment where one astronaut is selected at random from two different
crews and the sum of their isolation tolerance scores is observed. Construct a
probability distribution for this experiment.
#4
Consider an experiment where one astronaut is selected at random from two different
crews and the product of their isolation tolerance scores is observed. Construct a
probability distribution for this experiment.
#1 S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}
#2
X
P(X)
1
X
P(X)
1
0
/25
1
2
/49
2
1
/25
2
2
/25
3
3
/49
3
4
/49
4
3
/25
2
4
/5
5
/49
5
5
/25
8
#3
6
/49
7
/49
6
8
/7
1
6
9
/49
10
/49
5
11
/49
4
12
/49
3
13
/49
2
#4
X
P(X)
15
2
/49
1
/49
16
1
/49
1
2
/49
18
2
/49
2
3
/49
20
2
/49
2
4
/49
21
2
/49
3
5
/49
24
2
/49
2
6
/49
25
1
/49
4
7
/49
28
2
/49
2
8
/49
30
2
/49
2
9
/49
35
2
/49
1
10
/49
36
1
/49
2
12
/49
42
2
/49
4
14
/49
49
1
/49
2
14
/49
1
Assignment 3.4
Problems
#1
A twelve-sided die is tossed, find the probability of rolling a number greater than eight.
#2
A pair of six-sided dice are tossed. Find the probability of double.
#3
An experiment requires the toss of an eight-sided die and a six-sided die. Identify the
sample space for the experiment.
#4
An experiment requires observing the sum of the results of the toss of an eight-sided die
and a six-sided die. Construct a probability distribution for the experiment.
#5
An experiment requires observing the product of the results of the toss of an eight-sided
die and a six-sided die. Construct a probability distribution for the experiment.