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ELEMENTARY STATISTICS 3E
William Navidi and Barry Monk
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
Basic Concepts in Probability
Section 5.1
Objectives
1.
2.
3.
4.
Construct sample spaces
Compute and interpret probabilities
Approximate probabilities using the Empirical Method
Approximate probabilities by using simulation
Objective 1
Construct sample spaces
Probability
A probability experiment is one in which we do not know what any
individual outcome will be, but we do know how a long series of
repetitions will come out.
For example, if we toss a fair coin, we do not know what the
outcome of a single toss will be, but we do know what the outcome
of a long series of tosses will be – about half “heads” and half
“tails”.
The probability of an event is the proportion of times that the event
occurs in the long run. So, for a “fair” coin, that is, one that is
equally likely to come up heads as tails, the probability of heads is
1/2 and the probability of tails is 1/2.
Law of Large Numbers
The Law of Large Numbers says that as a probability
experiment is repeated again and again, the proportion of
times that a given event occurs will approach its
probability.
Sample Space
The collection of all the possible outcomes of a probability
experiment is called a sample space.
Example:
Describe the sample space for each of the following:
a) The toss of a coin
The sample space is {Heads, Tails}.
b) The roll of a die
The sample space is {1, 2, 3, 4, 5, 6}
c) The selection of a student at random from a list of 10,000
at a large university
The sample space consists of the 10,000 students.
Probability Model
We are often concerned with occurrences that consist of several
outcomes. For example, when rolling a die, we might be interested
in the probability of rolling an odd number. Rolling an odd number
corresponds to the collection of outcomes {1, 3, 5} from the sample
space {1, 2, 3, 4, 5, 6}. In general, a collection of outcomes of a
sample space is called an event.
Once we have a sample space, we need to specify a probability of
each event. This is done with a probability model. We use the letter
“ ” to denote probabilities. For example, we denote the probability
In general, if A denotes an event, the probability of event A is
denoted by (A).
Objective 2
Compute and interpret probabilities
Probability Rules
The probability of an event is always between 0 and 1. In
other words, for any event A, 0 ≤ (A) ≤ 1.
• If A cannot occur, then (A) = 0.
• If A is certain to occur, then (A) = 1.
Probabilities with Equally Likely Outcomes
If a sample space has equally likely outcomes and an event A has
Number of outcomes in A

outcomes, then (A) =
=
Number of outcomes in the sample space
Example:
In the Georgia Cash-4 Lottery, a winning number between 0000 and
9999 is chosen at random, with all the possible numbers being equally
likely. What is the probability that all four digits are the same?
The outcomes in the sample space are the numbers from 0000 to
9999, so there are 10,000 equally likely outcomes in the sample space.
There are 10 outcomes for which all the digits are the same: 0000,
1111, 2222, and so on up to 9999.
(All four digits the same) =
10
10,000
= 0.001
Sampling is a Probability Experiment
Sampling an individual from a population is a probability
experiment. The population is the sample space and
members of the population are equally likely outcomes.
Example: Sampling as a Probability Experiment
There 10,000 families in a certain town categorized as follows:
Own a house
Own a condo
Rent a house
Rent an apartment
4753
1478
912
2857
A pollster samples a single family from this population.
a) What is the probability that the sampled family owns a house?
b) What is the probability that the sampled family rents?
Solution:
a) The sample space consists of 10,000 households. Of these, 4753
4753
own a house. Therefore, (Own a house) =
= 0.4753.
10,000
b) The number of families who rent is 912 + 2857 = 3769. Therefore,
3769
(Rents) =
= 0.3769.
10,000
Unusual Events
An unusual event is one that is not likely to happen. In other words, an
event whose probability is small. There are no hard-and-fast rules as to
just how small a probability needs to be before an event is considered
unusual, but we will use the following rule of thumb.
Any event whose probability is less than 0.05 is considered to be
unusual.
Example:
In a college of 5000 students, 150 are math majors. A student is
selected at random and turns out to be a math major. Is this unusual?
Solution:
The event of choosing a math major consists of 150 students out of a
150
total of 5000. The probability of choosing a math major is
= 0.03.
5000
This would be considered an unusual event.
Objective 3
Approximate probabilities using the Empirical Method
Approximating Probabilities – Empirical Method
The Law of Large Numbers says that if we repeat a
probability experiment a large number of times, then the
proportion of times that a particular outcome occurs is likely
to be close to the true probability of the outcome.
The Empirical Method consists of repeating an experiment a
large number of times, and using the proportion of times an
outcome occurs to approximate the probability of the
outcome.
Example: Empirical Method
In a recent year, there were 2,046,935 boys and 1,952,451 girls
born in the U.S. Approximate the probability that a newborn baby
is a boy.
Solution:
We compute the number of times the experiment has been
repeated:
2,046,935 + 1,952,451 = 3,999,386 births.
The proportion of births that are boys is
approximate (Boy) ≈ 0.5118.
2,046,935
3,999,386
= 0.5118. We
Objective 4
Approximate probabilities by using simulation
Simulation
In practice, it can be difficult or impossible to repeat an
experiment many times in order to approximate a
probability with the Empirical Method. In some cases, we
can use technology to repeat an equivalent virtual
experiment many times. Conducting a virtual experiment in
this way is called simulation.
Simulation on the TI-84 PLUS
The randInt command on the TI-84 PLUS calculator may
be used to simulate the rolling of a single die. To access
this command, press MATH, scroll to the PRB menu, and
select randInt. The following screenshots illustrate how
to simulate the rolling of a die 100 times. The outcomes
are stored in list L1.
Simulation in EXCEL
Suppose that three dice are rolled, the smallest possible total is 3 and the largest possible
total is 18. Technology, such as EXCEL, can be used to “roll” the dice as many times as
desired. Following are the frequencies for each possible outcome when the dice are “rolled”
1000 times. We may, for example, estimate the probability of obtaining the number 10 as
128
(10) = 1000 = 0.128.
You Should Know . . .
• The Law of Large Numbers
• The definition of:
– Probability
– Sample space
– Event
• How to compute probabilities with equally likely
outcomes
• The rule-of-thumb for determining when an event A is
uncovered long-term spaces that cost \$1.50 per hour, and 57 are
uncovered short-term spaces that cost \$4.00 per hour. A parking
space is selected at random. Let represent the hourly parking
fee for the randomly sampled space. Find the probability
distribution of .
Solution:
To find the probability distribution, we must list the possible
values of and then find the probability of each of them. The
possible values of are 1.50, 2.00, 4.00, 4.50. Next, we find their
probabilities.
Example: Connection with Populations (Continued)
An airport parking facility contains 1000 parking spaces. Of these, 142 are
covered long-term spaces that cost \$2.00 per hour, 378 are covered shortterm spaces that cost \$4.50 per hour, 423 are uncovered long-term spaces
that cost \$1.50 per hour, and 57 are uncovered short-term spaces that cost
\$4.00 per hour.
1.50
2.00
4.00
4.50
# of spaces costing \$1.50
423
=
=
= 0.423
total # of spaces
1000
# of spaces costing \$2.00
142
=
=
= 0.142
total #of spaces
1000
# of spaces costing \$4.00
57
=
=
= 0.057
total # of spaces
1000
# of spaces costing \$4.50
378
=
=
= 0.378
total # of spaces
1000
x
1.50
2.00
4.00
4.50
P(x)
0.423
0.142
0.057
0.378
ҧ is a Random Variable
Often when we draw a s …