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Chapter 8
Probability
Chapter 8


Probability is the numerical likelihood,
measured between 0 and 1, that an uncertain
event will occur.
Probabilities specify the chance of an event
happening.
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Chapter 8

First property of probability: The probability
that an uncertain event will occur is between 0
and 1:
0 ≤ 𝑃 𝐸𝑖 ≤ 1
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Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Second property of probability: If 𝐸𝑖 is an
event representing some element in a sample
space, then:
𝑃 𝐸𝑖 = 1
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Three generally accepted ways to approach
probability:

Relative frequency approach
# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡 ℎ𝑎𝑠 𝑜𝑐𝑐𝑢𝑟𝑒𝑑
𝑃 𝐸𝑖 =
𝑇𝑜𝑡𝑎𝑙 # 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Three generally accepted ways to approach
probability:


Subjective approach
Classical approach
# 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑃 𝐸𝑖 =
𝑇𝑜𝑡𝑎𝑙 # 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
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Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8
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Relationships are important
Mutually exclusive events. Events that cannot occur
jointly; if one event happens, the other cannot.
Collectively exhaustive events. The set of all possible
outcomes of an experiment (i.e. the sample space).
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Combined probability of collectively exhaustive events
equal 1.
𝑃 1 𝑜𝑟 2 𝑜𝑟 3 𝑜𝑟 4 𝑜𝑟 5 𝑜𝑟 6 = 1
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Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8


Independent events. Two events are independent if
the occurrence of one event has no effect on the
probability that the second will occur.
Complementary events. Two events are
complementary if the failure of one to occur means the
other must occur. Complementary events are
collectively exhaustive.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Unions, Intersections, & Venn Diagrams.
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𝐴∪𝐵
𝐴∩𝐵
Venn diagram.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8
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Joint probability. Probability that two or more
events will occur.
Marginal probability. Probability of
occurrence of a single event.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Rule of multiplication used to determine joint
probability of A and B.
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If A and B are independent events, multiply probability of A
by probability of B.
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) × 𝑃(𝐵)

If A and B are dependent events, multiply probability of A
by probability of B, given B has already occurred.
𝑃 𝐴𝐵
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Rule of multiplication used to determine joint
probability of A and B.

If A and B are dependent events, multiply probability of A
by probability of B, given B has already occurred.
𝑃 𝐴𝐵
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Conditional probability. The probability that some event
will occur given that another event has already occurred.
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴𝐵 =
𝑃(𝐵)
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Rule of addition used to determine joint probability of
A or B.
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If A and B are mutually exclusive events, add probability of
A to probability of B.
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
If A and B are not mutually exclusive events, add probability
of A and probability of B, and subtract joint probability of A
and B.
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Double counting. Counting a single event twice.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Counting techniques. Methods to determine how
many subsets can be obtained from a set of objects.
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Permutations. A set of items in which both composition
and order are important.
Combinations. The set of items in which only composition
is important (order makes no difference).
Multiple-choice arrangements. Order makes a difference
but, duplication of individual values is allowed.
Multiplication method. Used when choices must be made
from two or more distinct groups.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Permutations.
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How many permutations are possible given the set, ABC?
nPr =
𝑛!
𝑛−𝑟 !
n = number of elements
r = number of elements at a time
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Combinations.
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How many combinations are possible given n = 10, r = 3?
nCr =
𝑛!
𝑟! 𝑛−𝑟 !
n = number of elements
r = number of elements at a time
Given any set of conditions involving the selection of r items
from a list of n items, where 𝑟 ≤ 𝑛, the number of possible
permutations will exceed the number of combinations.
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8
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Random variable. A variable whose outcome occurs by
chance (number of units sold, daily output, height of
students…)
Probability distribution. A list of all possible outcomes of an
experiment and the probabilities associated with each outcome
presented in the form of a table, graph, or formula.
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Uniform probability distribution. The probabilities of all outcomes are
the same.
Discrete probability distribution. The variable can take on only certain
values (usually whole numbers).
Continuous probability distribution. The variable can take on an infinite
number of values (dependent on the precision of the measuring tool).
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Binomial distribution. Each trial results in one of only two
mutually exclusive outcomes, one is identified as a success the
other, failure. The probability of each outcome remains
constant from one trial to the next.
𝑛!
𝑃 𝑥 =
𝜋 𝑥 (1 − 𝜋)𝑛−𝑥
𝑥! 𝑛 − 𝑥 !
π = probability of success
n = sample size
x = value of interest (defects, failures)
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Hypergeometric distribution. Used if the probability
of success is not constant ( population is small, sample
contains a large portion of the population).
𝑃 𝑥 =
rCx N−rCn−x
NCn
N = population size
n = sample size
r = number in the population identified as a success
x = number in the sample identified as a success
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Donna C. S. Summers
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Poisson distribution. Measures the probability of a random
event over some interval of time or space (number of customers
per hour, accidents per month, number of defective electrical
connections per mile of wiring…).
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Probability of the occurrence of the event is constant for any two
intervals of time or space.
Occurrence of the event in any interval is independent of the occurrence
in any other interval.
𝜇 𝑥 𝑒 −𝜇
𝑃 𝑥 =
𝑥!
x = number of times the event occurs
𝜇 = mean number of occurrence per unit of time or space
𝑒 = 2.71828 (base of the natural log system)
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8
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Normal distribution.
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Continuous distribution
Variables such as height, weight, distance
Variables generally the result of measurements
Shape and positon determined by 𝜇 and 𝜎
𝑋𝑖 − 𝑋
𝑍=
𝑠
𝑋𝑖 = some specified value for the variable/value of interest
𝑋 = average
𝑠 = standard deviation
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.
Chapter 8

Questions?
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© 2010 Pearson Higher Education,
Upper Saddle River, NJ 07458. • All Rights Reserved.