Transcript Slide 1

13
THE NATURE
OF PROBABILITY
Copyright © Cengage Learning. All rights reserved.
13.1
Introduction to Probability
Copyright © Cengage Learning. All rights reserved.
Terminology
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Terminology
An experiment is an observation of any physical
occurrence. The sample space of an experiment is the set
of all its possible outcomes.
An event is a subset of the sample space.
If an event is the empty set, it is called the impossible
event; and if it has only one element, it is called a simple
event.
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Example 1 – Find a sample space
Suppose a researcher wishes to study how the color of a
child’s eyes is related to the parents’ eye color.
a. List the sample space.
b. Give an example of a simple event.
c. Give an example of an impossible event.
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Example 1 – Solution
a. The sample space is a listing of all possible eye colors:
{green, blue, brown, hazel}.
b. A simple event is an event with only one element, say
{blue}.
c. An impossible event is one that is empty, say, {purple}.
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Terminology
Die is a cube showing six faces marked 1, 2, 3, 4, 5, and 6.
If an outcome of an experiment has the same chance of
occurring as any other outcome, then they are said to be
equally likely outcomes.
Two events E and F are said to be mutually exclusive
if E  F =
that is, they cannot both occur simultaneously.
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Probability
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Probability
A probabilistic model deals with situations that are
random in character and attempts to predict the outcomes
of events with a certain stated or known degree of
accuracy.
For example, if we toss a coin, it is impossible to predict in
advance whether the outcome will be a head or a tail.
Our intuition tells us that it is equally likely to be a head or a
tail, and somehow we sense that if we repeat the
experiment of tossing a coin a large number of times, a
head will occur “about half the time.”
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Probability
To check this out, I recently flipped a coin 1,000 times and
obtained 460 heads and 540 tails. The percentage of
heads is
= 0.46 = 46%, which is called the relative
frequency.
If an experiment is repeated n times and an event occurs m
times, then
is called the relative frequency of the event
Our task is to create a model that will assign a number p,
called the probability of an event, which will predict the
relative frequency.
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Probability
This means that for a sufficiently large number of
repetitions of an experiment,
Probabilities can be obtained in one of three ways:
1. Empirical probabilities (also called a posteriori models)
are obtained from experimental data. For example, an
assembly line producing brake assemblies for General
Motors produces 1,500 items per day.
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Probability
The probability of a defective brake can be obtained by
experimentation. Suppose the 1,500 brakes are tested
and 3 are found to be defective.
Then the empirical probability is the relative frequency of
occurrence, namely,
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Probability
2. Theoretical probabilities (also called a priori models)
are obtained by logical reasoning according to stated
definitions.
For example, the probability of rolling a die and obtaining
a 3 is because there are six possible outcomes, each
with an equal chance of occurring, so a 3 should appear
of the time.
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Probability
3. Subjective probabilities are obtained by experience
and indicate a measure of “certainty” on the part of the
speaker. These probabilities are not necessarily arrived
at through experimentation or theory.
For example, a dad says to his daughter, “After that
outburst the probability that you will go to the dance is
almost 0%.”
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Probability
Our focus is on theoretical probabilities, but we should
keep in mind that our theoretical model should be
predictive of the results obtained by experimentation
(empirical probabilities); if they are not consistent, and we
have been careful about our record keeping in arriving at
an empirical probability, we would conclude that our
theoretical model is faulty.
The reason for this conclusion is called the law of large
numbers.
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Probability
This law of large numbers keeps us “honest” in maintaining
our records and in setting up models for finding the
theoretical probability.
The simplest probability model makes certain assumptions
about the sample space.
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Probability
If the sample space can be divided into mutually exclusive
and equally likely outcomes, we can define the probability
of an event.
Let’s consider the experiment of tossing a single coin. A
suitable sample space is
S = {heads, tails}
Suppose we wish to consider the event of obtaining heads;
we’ll call this event A. Then
A = {heads}
and this is a simple event.
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Probability
We wish to define the probability of event A, which we
denote by P(A).
Notice that the outcomes in the sample space are mutually
exclusive; that is, if one occurs, the other cannot occur.
If we flip a coin, there are two possible outcomes, and one
and only one outcome can occur on a toss.
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Probability
If each outcome in the sample space is equally likely, we
define the probability of A as
A “successful” result is a result that corresponds to the
event whose probability we are seeking—in this case,
{heads}.
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Probability
Since we can obtain a head (success) in only one way, and
the total number of possible outcomes is two, the
probability of heads is given by this definition as
P(heads) = P(A) =
This must correspond to the empirical results you would
obtain if you repeated the experiment a large number of
times.
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Probability
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Example 5 – Find the probability of drawing from an urn
Consider a jar that contains marbles as shown in
Figure 13.3. Suppose that each marble has an equal
chance of being picked from the jar. Find:
a. P(blue)
b. P(green)
c. P(yellow)
Marbles in jar
Figure 13.3
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Example 5 – Solution
a. P(blue) =
4 blue marbles in jar
12 blue marbles in jar
=
Reduce fractions
b. P(green) =
c. P(yellow) =
This is a theoretical probability; also note that the
probabilities of all the simple events sum to
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Probability
Let us find the smallest and largest probabilities of an
event.
The probability of the empty set is 0, which means that the
event cannot occur. In the problem set, you will be asked to
show that the probability of an event that must occur is 1.
These are the two extremes. All other probabilities fall
somewhere in between.
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Probability
The closer a probability is to 1, the more likely the event is
to occur; the closer a probability is to 0, the less likely the
event is to occur.
If the probability you are finding has a sample space that is
easy to count, and if all simple events in the sample space
are equally likely, then there is a simple procedure that we
can use to find probabilities.
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Probability
Keep in mind that this procedure does not apply to every
situation. If it doesn’t, you need a more complicated model,
or else you must proceed experimentally.
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Probabilities of Unions and
Intersections
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Probabilities of Unions and Intersections
The word or is translated as  (union), and the word and is
translated as  (intersection).
We will find the probabilities of compound events involving
the words or and and by finding unions and intersections of
events.
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Example 11 – Comparing or and and
Suppose that a single card is selected from an ordinary
deck of cards.
a. What is the probability that it is a two or a king? (See
Figure 13.6a.)
b. What is the probability that it is a two or a heart? (See
Figure 13.6b.)
Finding Probabilities.
Figure 13.6
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Example 11 – Comparing or and and
cont’d
c. What is the probability that it is a two and a heart?
d. What is the probability that it is a two and a king?
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Example 11 – Solution
a. P(two or a king) = P(two  king).
Look at Figure 12.2. if you are not familiar with a
deck of cards.
A deck of cards
Figure 12.2
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Example 11 – Solution
cont’d
two = {two of hearts, two of spades, two of diamonds,
two of clubs}
king = {king of hearts, king of spades, king of diamonds,
king of clubs}
two  king = {two of hearts, two of spades, two of
diamonds, two of clubs, king of hearts, king
of spades, king of diamonds, king of clubs}
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Example 11 – Solution
cont’d
There are 8 possibilities for success. It is usually not
necessary to list all of these possibilities to know that
there are 8 possibilities (four twos and four kings):
P(two  king) =
b. This seems to be very similar to part a, but there is one
important difference. Look at the sample space and
notice that although there are 4 twos and 13 hearts, the
total number of successes is not 4 + 13 = 17, but rather 16.
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Example 11 – Solution
cont’d
two = {two of hearts, two of spades, two of diamonds,
two of clubs}
heart = {ace of hearts, two of hearts, three of
hearts,…, king of hearts}
two  heart = {two of hearts, two of spades, two of
diamonds, two of clubs, ace of hearts,
three of hearts, four of hearts, five of
hearts, six of hearts, seven of hearts,
eight of hearts, nine of hearts, ten of
hearts, jack of hearts, queen of hearts,
king of hearts}.
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Example 11 – Solution
cont’d
It is not necessary to list these possibilities. The purpose
of doing so in this case was to reinforce the fact that
there are actually 16 (not 17) possibilities.
The cardinality of a union:
= 4 + 13 – 1
= 16
Thus we see
P(two  heart) =
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Example 11 – Solution
cont’d
c. For this part, we seek two  heart = {two of hearts} so
there is one element in the intersection:
P(two  heart) =
d. Finally, two  king =
intersection:
so there are no elements in the
P(two  king) =
= 0.
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