Section 5-1
Review and Preview
Review and Preview
This chapter combines the methods of descriptive
statistics presented in Chapter 2 and 3 and those of
probability presented in Chapter 4 to describe and
analyze probability distributions.
Probability Distributions describe what will probably
happen instead of what actually did happen, and they
are often given in the format of a graph, table, or
formula.
Preview
In order to fully understand probability distributions,
we must first understand the concept of a random
variable, and be able to distinguish between discrete
and continuous random variables. In this chapter we
focus on discrete probability distributions. In
particular, we discuss binomial probability
distributions.
Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions by
presenting possible outcomes along with the relative frequencies we
expect.
Section 5-2
Random Variables
Random Variable
Probability Distribution
Random variable: a variable (typically represented
by x) that has a single numerical value, determined by
chance, for each outcome of a procedure. (Example:
the number of peas with green pods among 5
offspring peas.)
Probability distribution: a description that gives the
probability for each value of the random variable;
often expressed in the format of a graph, table, or
formula.
Discrete and Continuous Random Variables
Discrete random variable: either a finite number of
values or countable number of values, where
“countable” refers to the fact that there might be
infinitely many values, but they result from a
counting process. (it cannot be a decimal)
Continuous random variable: infinitely many values,
and those values can be associated with
measurements on a continuous scale without gaps or
interruptions. (if something could be a decimal,
it is continuous)
Example 1: Identify the given random variable as
being discrete or continuous.
a) The number of people now driving a car in the
United States.
Discrete
b) The weight of the gold stored in Fort Knox.
Continuous
c) The height of the last airplane departed from JFK
Airport in New York City.
Continuous
Example 1 continued: Identify the given random
variable as being discrete or continuous.
d) The number of cars in San Francisco that crashed last
year.
Discrete
e) The time required to fly from Los Angeles to
Shanghai.
Continuous
Requirements for
Probability Distribution
The sum of all probabilities is 1.
ΣP(x) = 1, where x assumes all possible values.
(Values such as 0.999 or 1.001 are acceptable due to roundoff
error)
Each individual probability is a value between 0 and
1 inclusive.
0  P(x)  1, for every individual value of x.
Mean, Variance and
Standard Deviation of a
Probability Distribution
µ = Σ [x • P(x)]
Mean
2
σ = Σ[(x – µ)2 • P(x)]
2
2
σ = Σ[x • P(x)] – µ
σ=
2
Σ[x2 • P(x)] – µ2
Variance
Variance (shortcut)
Standard Deviation
Roundoff Rule for µ, , and 
2
Round results by carrying one more decimal place
than the number of decimal places used for the
random variable x.
If the values of x are integers, round µ, σ, and σ2 to
one decimal place.
Example 2: Determine whether or not a probability distribution is
given. If a probability distribution is given, find its mean and
standard deviation. If a probability distribution is not given, identify
the requirement(s) that are not satisfied. Three males with X-linked
genetic disorder have one child each. The random variable x is the
number of children among the three who inherit the X-linked genetic
disorder.
x
P(x)
0
0.125
The given table is a probability distribution since
0 ≤ P(x) ≤ 1 for each x and ΣP(x) = 1.
1
0.375
2
0.375
    x  P( x)
3
0.125
 1.5 children
 2    x 2  P( x)    2
 3.000  1.5 
2
 0.750
  0.866, rounded to 0.9 children
Example 3: Determine whether or not a probability distribution is
given. If a probability distribution is given, find its mean and
standard deviation. If a probability distribution is not given, identify
the requirement(s) that are not satisfied. Air America has a policy of
routinely overbooking flights. The random variable x represents the
number of passengers who cannot be boarded because there are more
passengers than seats (based on data from an IBM research paper by
Lawrence, Hong, and Cherrier.)
The given table is not a probability distribution
since ΣP(x) = 0.984 ≠ 1.
x
P(x)
0
0.051
1
0.141
2
0.274
3
0.331
4
0.187
Example 4: Determine whether or not a probability distribution is
given. If a probability distribution is given, find its mean and
standard deviation. If a probability distribution is not given, identify
the requirement(s) that is/are not satisfied.
x
P(x)
The given table is not a probability distribution
since ΣP(x) = 1.2 ≠ 1.
1
0.6
2
0.2
3
0.2
4
0.15
5
0.05
Identifying Unusual Results
Range Rule of Thumb
According to the range rule of thumb, most
values should lie within 2 standard deviations of
the mean.
We can therefore identify “unusual” values by
determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
Identifying Unusual Results
Probabilities
Rare Event Rule for Inferential Statistics
If, under a given assumption (such as the
assumption that a coin is fair), the probability of a
particular observed event (such as 992 heads in
1000 tosses of a coin) is extremely small, we
conclude that the assumption is probably not
correct.
Identifying Unusual Results
Probabilities
Using Probabilities to Determine When Results
Are Unusual
Unusually high: x successes among n trials is an
unusually high number of successes if
P(x or more) ≤ 0.05.
Unusually low: x successes among n trials is an
unusually low number of successes if
P(x or fewer) ≤ 0.05.
Example 5: Refer to the table, which describes results
from eight offspring peas. The random variable x
represents the number of offspring peas with green
pods.
x
P(x)
a) Find the probability of getting exactly
7 peas with green pods.
b) Find the probability of getting 7 or
more peas with green pods.
0
0+
1
0+
2
0.004
3
0.023
4
0.087
5
0.208
6
0.311
7
0.267
8
0.100
Example 5 continued: Refer to the table, which
describes results from eight offspring peas. The
random variable x represents the number of offspring
peas with green pods.
x
P(x)
c) Which probability is relevant for
determining whether 7 is an unusually
high number of peas with green pods: the
result from part (a) or part (b)?
d) Is 7 an unusually high number of peas
with green pods? Why or why not?
0
0+
1
0+
2
0.004
3
0.023
4
0.087
5
0.208
6
0.311
7
0.267
8
0.100
Example 6: Based on past results found in the
Information Please Almanac, there is a 0.1919
probability that a baseball World Series contest will last
four games, a 0.2121 probability that it will last five
games, a 0.2222 probability that it will last six games,
and a 0.3737 probability that it will last seven games.
a) Does the given information describe a probability
distribution?
Example 6 continued: Based on past results found in
the Information Please Almanac, there is a 0.1919
probability that a baseball World Series contest will last
four games, a 0.2121 probability that it will last five
games, a 0.2222 probability that it will last six games,
and a 0.3737 probability that it will last seven games.
b) Assuming that the given information describes a
probability distribution, find the mean and standard
deviation for the numbers of games in World Series
contests.
Example 6 continued: Based on past results found in
the Information Please Almanac, there is a 0.1919
probability that a baseball World Series contest will last
four games, a 0.2121 probability that it will last five
games, a 0.2222 probability that it will last six games,
and a 0.3737 probability that it will last seven games.
c) Is it unusual for a team to “sweep” by winning in
four games? Why or why not?
Example 7: Based on data from CarMax.com, when a
car is randomly selected, the number of bumper stickers
and the corresponding probabilities are as follows:
0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012);
5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9(0.004).
a) Does the given information describe a probability
distribution?
Example 7 continued: Based on data from
CarMax.com, when a car is randomly selected, the
number of bumper stickers and the corresponding
probabilities are as follows: 0 (0.824); 1 (0.083);
2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008);
7 (0.004); 8 (0.004); 9(0.004).
b) Assuming that a probability distribution is described,
find its mean and standard deviation.
Example 7 continued: Based on data from
CarMax.com, when a car is randomly selected, the
number of bumper stickers and the corresponding
probabilities are as follows: 0 (0.824); 1 (0.083);
2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008);
7 (0.004); 8 (0.004); 9(0.004).
c) Use the range rule of thumb to identify the range of
values for usual numbers of bumper stickers.
Example 7 continued: Based on data from
CarMax.com, when a car is randomly selected, the
number of bumper stickers and the corresponding
probabilities are as follows: 0 (0.824); 1 (0.083);
2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008);
7 (0.004); 8 (0.004); 9(0.004).
d) Is it unusual for a car to have more than one bumper
sticker? Why or why not?
Example 8: Let the random variable x represent the number of
girls in a family of three children. Construct a table describing the
probability distribution, then find the mean and standard
deviation. (Hint: List the different possible outcomes) Is it unusual
for a family of three children to consist of three girls?
Expected Value
The expected value of a discrete random
variable is denoted by E, and it represents the
mean value of the outcomes. It is obtained by
finding the value of Σ [x • P(x)].
E = Σ[x • P(x)]
Example 9: In the Illinois Pick 3 lottery game, you pay
50¢ to select a sequence of three digits, such as 233. If
you select the same sequence of three digits that are
drawn, you win and collect $250.
a) How many different selections are possible?
b) What is the probability of winning?
c) If you win, what is your net profit?
Example 9 continued: In the Illinois Pick 3 lottery
game, you pay 50¢ to select a sequence of three digits,
such as 233. If you select the same sequence of three
digits that are drawn, you win and collect $250.
d) Find the expected value.
e) If you bet 50 ¢ in Illinois’ Pick 4 game, the expected
value is –25¢. Which bet is better: A 50¢ bet in the
Illinois Pick 3 game or a 50¢ bet in the Illinois Pick 4
game? Explain.