Excursions in Modern
Mathematics
Sixth Edition
Peter Tannenbaum
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Chapter 15
Chances, Probabilities, and Odds
Measuring
Uncertainty
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Chances, Probabilities, and Odds
Outline/learning Objectives
To describe an appropriate sample space
of a random experiment.
 To apply the multiplication rule,
permutations, and combinations to
counting problems.
 To understand the concept of a
probability assignment.

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Chances, Probabilities, and Odds
Outline/learning Objectives
To identify independent events and their
properties.
 To use the language of odds in
describing probabilities of events.
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Chances, Probabilities, and Odds
15.1 Random Experiments
and Sample Spaces
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Chances, Probabilities, and Odds


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Random experiment
Description of an activity or process whose
outcome cannot be predicted ahead of time.
Sample space
Associated with every random experiment is the
set of all of its possible outcomes. We will
consistently use the letter S to denote a sample
space and N to denote its size (the number of
outcomes in S).
Chances, Probabilities, and Odds
Rolling the Dice: Part 1
One of the most common things we do with dice is to roll
a pair of dice and consider just the total of the the two
die. A more general scenario is when we do care what
number each individual turns up. Here below we have a
sample space with 36 different outcomes.
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Chances, Probabilities, and Odds
Rolling the Dice: Part 1
When looking at the figure below you will notice that we
are treating the dice as distinguishable objects (as if one
were white and the other red), so that
and
are considered different outcomes.
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Chances, Probabilities, and Odds
15.2 Counting Sample
Spaces
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Chances, Probabilities, and Odds
The Multiplication Rule
When something is done in stages, the
number of ways it can be done is found
by multiplying the number of ways each
of the stages can be done.
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Chances, Probabilities, and Odds
The Making of a Wardrobe: Part 2
Our strategy will be to think of an outfit as being put together in
stages and to draw a box for each of the stages. We then
separately count the number of choices at each stage and
enter that number in the corresponding box.
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Chances, Probabilities, and Odds
The Making of a Wardrobe: Part 2
The last step is to multiply the numbers in each box. The final
count for the number of different outfits is
N = 3  7  27  3 = 1701
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Chances, Probabilities, and Odds
15.3 Permutations
and Combinations
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Chances, Probabilities, and Odds
Permutation
A group of objects where the ordering of the
objects within the group makes a difference.
 Combination
A group of objects in which the ordering of the
objects is irrelevant.

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Chances, Probabilities, and Odds
The Pleasures of Ice Cream: Part 1
Say you want a true double in a bowl – how many different
choices so you have?
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Chances, Probabilities, and Odds
The Pleasures of Ice Cream: Part 1
The natural impulse is to count the number of choices using the
multiplication rule (and a box model) as shown below. This
would give an answer of 930.
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Chances, Probabilities, and Odds
The Pleasures of Ice Cream: Part 1
Unfortunately, this answer is double counting each of the true
doubles. Why?
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Chances, Probabilities, and Odds
The Pleasures of Ice Cream: Part 1
When we use the multiplication rule, there is a well-defined
order to things, and a scoop of strawberry followed by a
scoop of chocolate is counted separate from a scoop of
chocolate followed by a scoop of strawberry.
The good news is that now we understand why the count of 930
is wrong and we can fix it. All we have to do is divide the
original count by 2.
(31  30)/2 = 465
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Chances, Probabilities, and Odds
15.4 Probability
Spaces
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Chances, Probabilities, and Odds
Event
Any subset of the sample space.

Simple event
An event that consists of just one outcome.
 Impossible event
A special case of the empty set { },
corresponding to an event with no outcomes.

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Chances, Probabilities, and Odds
Probability assignment
A function that assigns to each event E a number
between 0 and 1, which represents the
probability of the event E and which we denote
by Pr (E).
 Probability space
Once a specific probability assignment is made
on a sample space, the combination of the
sample space and the probability assignment.

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Chances, Probabilities, and Odds
Elements of a Probability Space
 Sample space: S = {o1, o2,…., oN}
 Probability assignment: Pr(o1),Pr(o2),… Pr(oN)
[Each of these is a number between 0 and 1 satisfying
Pr(o1) + Pr(o2) + … Pr(oN) = 1]
 Events: These are all the subsets of S, including { }
and S itself. The probability of an event is given by the
sum of the probabilities of the individual outcomes that
make up the event. [In particular, Pr({ }) = 0 and
Pr(S) =1]
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Chances, Probabilities, and Odds
15.5 Equiprobable
Spaces
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Chances, Probabilities, and Odds
Probabilities in Equiprobable Spaces
Pr(E) = k/N (where k denotes the size of the
event E and N denotes the size of the sample
space S).
A probability space where each simple event has
an equal probability is called an equiprobable
“equal opportunity” space.
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Chances, Probabilities, and Odds
Rolling the Dice: Part 2
The sample space has N = 36 individual outcomes, each with
probability 1/36. We will use the notation T2, T3, …T12 to
describe the events “roll a total of 2,” “roll a total of 3,” …,
“roll a total of 12,” respectively. We show you how to find
Pr(T7) and Pr(T11),
T11 =
, Thus,
Pr(T11) = 2/36  0.056
T7 =
25
, Thus,
Pr(T7) = 6/36 = 1/6  0.167
Chances, Probabilities, and Odds
Tallying
We can just write down all the individual
outcomes in the event E and tally their number.
This approach gives
and Pr(E) = 11/36.
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Chances, Probabilities, and Odds
Complementary Event
Imagine that you are playing a game, and you
win if at least one of the two numbers comes
up an Ace (that’s event E). Otherwise you lose
(call that event F). The two events E and F are
called complementary events. The
probabilities of complementary events add up
to 1. Thus,
Pr(E) = 1 – Pr(F).
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Chances, Probabilities, and Odds


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Independence Events
If the occurrence of one event does not affect the
probability of the occurrence of the the other.
Multiplication Principle for Independent Events
When events E and F are independent, the probability
that both occur is the product of their respective
probabilities; in other words,
Pr (E and F) = Pr(E) • Pr(F).
Chances, Probabilities, and Odds
15.6 Odds
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Chances, Probabilities, and Odds
Odds
Let E be an arbitrary event. If F denotes the
number of ways that event E can occur (the
favorable outcomes or hits), and U denotes the
number of ways that event E does not occur
(the unfavorable outcomes, or misses), then
the odds of (also called the odds in favor of),
the event E are given by the ratio F to U, and
the odds against the event E are given by the
ratio U to F.
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Chances, Probabilities, and Odds
Conclusion
 Sample
space
 Random experiment
 Events
 Probability assignment
 Equiprobable spaces
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