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15 Chances, Probabilities, and Odds
15.1 Random Experiments and Sample
Spaces
15.2 Counting Outcomes in Sample
Spaces
15.3 Permutations and Combinations
15.4 Probability Spaces
15.5 Equiprobable Spaces
15.6 Odds
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Excursions in Modern Mathematics, 7e: 15.1 - 1
Random Experiment
• Probability is the quantification of
uncertainty.
• We will use the term random
experiment to describe an activity or a
process whose outcome cannot be
predicted ahead of time.
• Examples of random experiments:
tossing a coin, rolling a pair of dice,
drawing cards out of a deck of cards,
predicting the result of a football game,
and forecasting the path of a hurricane.
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Excursions in Modern Mathematics, 7e: 15.1 - 2
Sample Space
• Associated with every random
experiment is the set of all of its possible
outcomes, called the sample space of
the experiment.
• For the sake of simplicity, we will
concentrate on experiments for which
there is only a finite set of outcomes,
although experiments with infinitely many
outcomes are both possible and
important.
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Excursions in Modern Mathematics, 7e: 15.1 - 3
Sample Space - Set Notation
• We use the letter S to denote a sample
space and the letter N to denote the size
of the sample space S (i.e., the number
of outcomes in S).
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Excursions in Modern Mathematics, 7e: 15.1 - 4
Example 15.1
Tossing a Coin
One simple random experiment is to toss a
quarter and observe whether it lands heads or
tails. The sample space can be described by
S = {H, T}, where H stands for Heads and T
for Tails. Here N = 2.
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Excursions in Modern Mathematics, 7e: 15.1 - 5
Example 15.2
More Coin Tossing
Suppose we toss a coin twice and record the
outcome of each toss (H or T) in the order it
happens. What is the sample space?
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Excursions in Modern Mathematics, 7e: 15.1 - 6
Example 15.2
More Coin Tossing
The sample space now is
S = {HH, HT, TH, TT}, where HT means that
the first toss came up H and the second toss
came up T, which is a different outcome from
TH (first toss T and second toss H). In this
sample space N = 4.
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Excursions in Modern Mathematics, 7e: 15.1 - 7
Example 15.2
More Coin Tossing
Suppose now we toss two distinguishable
coins (say, a nickel and a quarter) at the
same time. What is the sample space?
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Excursions in Modern Mathematics, 7e: 15.1 - 8
Example 15.2
More Coin Tossing
The sample space is still
S = {HH, HT, TH, TT}. (Here we must agree
what the order of the symbols is–for example,
the first symbol describes the quarter and the
second the nickel.)
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Excursions in Modern Mathematics, 7e: 15.1 - 9
Example 15.2
More Coin Tossing
Since they have the same sample space, we
will consider the two previous random
experiments as the same random experiment.
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Excursions in Modern Mathematics, 7e: 15.1 - 10
Example 15.2
More Coin Tossing
Suppose we toss a coin twice, but we only
care now about the number of heads that
come up. What is the sample space?
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Excursions in Modern Mathematics, 7e: 15.1 - 11
Example 15.2
More Coin Tossing
Here there are only three possible outcomes
(no heads, one head, or both heads), and
symbolically we might describe this sample
space as
S = {0, 1, 2}.
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Excursions in Modern Mathematics, 7e: 15.1 - 12
Example 15.5
Dice Rolling
The experiment is to roll a pair of dice. What
is the sample space?
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Excursions in Modern Mathematics, 7e: 15.1 - 13
Example 15.5
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More Dice Rolling
Excursions in Modern Mathematics, 7e: 15.1 - 14
Example 15.5
More Dice Rolling
Here we have a sample space with 36
different outcomes. Notice that the dice are
colored white and red, a symbolic way to
emphasize the fact that we are treating the
dice as distinguishable objects. That is why
the following rolls are distinguishable.
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Excursions in Modern Mathematics, 7e: 15.1 - 15
Example 15.5

More Dice Rolling
The sample space has 36 possible outcomes:
{(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)}
where the pairs represent the numbers rolled on
each dice (white, red).
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Excursions in Modern Mathematics, 7e: 15.1 - 16
Example 15.4
Rolling a Pair of Dice
Roll a pair of dice and consider the total of the
two numbers rolled. What is the sample
space?
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Excursions in Modern Mathematics, 7e: 15.1 - 17
Example 15.4
Rolling a Pair of Dice
The possible outcomes in this scenario range
from “rolling a two” to “rolling a twelve,” and
the sample space can be described by
S = {2, 3, 4, 5, 6, 7, 8, 9, 10,11,12}.
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Excursions in Modern Mathematics, 7e: 15.1 - 18
Examples
• Page 577, problem 3
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Excursions in Modern Mathematics, 7e: 15.1 - 19
Examples
• Page 577, problem 3
Solution:
• {ABCD, ABDC, ACBD, ACDB, ADBC,
ADCB, BACD, BADC, BCAD, BCDA,
BDAC, BDCA, CABD, CADB, CBAD,
CBDA, CDAB, CDBA, DABC, DACB,
DBAC, DBCA, DCAB, DCBA}
• There are 24 outcomes.
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Excursions in Modern Mathematics, 7e: 15.1 - 20
Not Listing All of the Outcomes
We would like to understand what the
sample space looks like without necessarily
writing all the outcomes down. Our real goal
is to find N, the size of the sample space. If
we can do it without having to list all the
outcomes, then so much the better.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 15.1 - 21
15 Chances, Probabilities, and Odds
15.1 Random Experiments and Sample
Spaces
15.2 Counting Outcomes in Sample
Spaces
15.3 Permutations and Combinations
15.4 Probability Spaces
15.5 Equiprobable Spaces
15.6 Odds
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Excursions in Modern Mathematics, 7e: 15.1 - 22
Example 15.7
Tossing More Coins
• If we toss a coin three times and
separately record the outcome of each
toss, the sample space is given by S =
{HHH, HHT, HTH, HTT, THH, THT, TTH,
TTT}.
• Here we can just count the outcomes and
get N = 8.
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Excursions in Modern Mathematics, 7e: 15.1 - 23
Example 15.7
Tossing More Coins
• Toss a coin 10 times
• In this case the sample space S is too big
to write down
• We can “count” the number of outcomes in
S without having to tally them one by one
using the multiplication rule.
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Excursions in Modern Mathematics, 7e: 15.1 - 24
Multiplication Rule
• Suppose an activity consists of a series of
events in which there are a possible
outcomes for the first event, b possible
outcomes for the second event, c possible
outcomes for the third event, and so on.
• Then the total number of different possible
outcomes for the series of events is:
a·b·c·…
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Excursions in Modern Mathematics, 7e: 15.1 - 25
Example 15.7
Tossing More Coins
• Toss a coin ten times. How many
outcomes are in the sample space?
• There are two outcomes on the first toss
• There are two outcomes on the second
toss, etc.
• The total number of possible outcomes is
found by multiplying ten two’s together.
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Excursions in Modern Mathematics, 7e: 15.1 - 26
Example 15.7
Tossing More Coins
• Total number of outcomes if a coin is
tossed ten times:
N
2
2
22
10 factors
• Thus N = 210 = 1024.
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Excursions in Modern Mathematics, 7e: 15.1 - 27
Example 15.8
The Making of a
Wardrobe
Dolores is a young saleswoman planning her
next business trip. She is thinking about
packing three different pairs of shoes, four
skirts, six blouses, and two jackets. How
many different outfits will she be able to
create by combining these items? (Assume
that an outfit consists of one pair of shoes,
one skirt, one blouse, and one jacket.)
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Excursions in Modern Mathematics, 7e: 15.1 - 28
Example 15.8
The Making of a
Wardrobe
Let’s assume that an outfit consists of one
pair of shoes, one skirt, one blouse, and one
jacket. Then to make an outfit Dolores must
choose a pair of shoes (three choices), a skirt
(four choices), a blouse (six choices), and a
jacket (two choices). By the multiplication rule
the total number of possible outfits is
3 4 6 2 = 144.
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Excursions in Modern Mathematics, 7e: 15.1 - 29
Example 15.10 Ranking the Candidate
in an Election: Part 2
Five candidates are running in an election,
with the top three vote getters elected (in
order) as President, Vice President, and
Secretary. How many different ways are there
to choose the five candidates to fill these
three positions?
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Excursions in Modern Mathematics, 7e: 15.1 - 30
Example 15.10 Ranking the Candidate
in an Election: Part 2
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Excursions in Modern Mathematics, 7e: 15.1 - 31
Examples
• Page 578, problem 14
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Excursions in Modern Mathematics, 7e: 15.1 - 32
Examples
• Solution to part (a)
8 7 6 5 4 3 2 1 40,320
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Excursions in Modern Mathematics, 7e: 15.1 - 33
Examples
• Solution to part (a)
4 7 6 5 4 3 2 1 20,160
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Excursions in Modern Mathematics, 7e: 15.1 - 34
Examples
• Solution to part (c)
4 4 3 3 2 2 1 1 576
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Excursions in Modern Mathematics, 7e: 15.1 - 35
15 Chances, Probabilities, and Odds
15.1 Random Experiments and Sample
Spaces
15.2 Counting Outcomes in Sample
Spaces
15.3 Permutations and Combinations
15.4 Probability Spaces
15.5 Equiprobable Spaces
15.6 Odds
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Excursions in Modern Mathematics, 7e: 15.1 - 36
Counting Problems
Many counting problems can be reduced to
a question of counting the number of ways
in which we can choose groups of objects
from a larger group of objects. Often these
problems require somewhat more
sophisticated counting methods than the
multiplication rule.
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Excursions in Modern Mathematics, 7e: 15.1 - 37
Counting Problems
In this section we will discuss the dual
concepts of permutation (a group of objects
in which the ordering of the objects within
the group makes a difference) and
combination (a group of objects in which the
ordering of the objects is irrelevant).
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Excursions in Modern Mathematics, 7e: 15.1 - 38
Permutation versus Combination
Suppose that we have a set of n distinct
objects and we want to select r different
objects from this set. The number of ways
that this can be done depends on whether
the selections are ordered or unordered.
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Excursions in Modern Mathematics, 7e: 15.1 - 39
Permutation versus Combination
To distinguish between these two scenarios,
we use the terms permutation to describe
an ordered selection and combination to
describe an unordered selection.
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Excursions in Modern Mathematics, 7e: 15.1 - 40
Permutation versus Combination
For a given number of objects n and a given
selection size r (where 0 ≤ r ≤ n) we can talk
about the “number of permutations of n
objects taken r at a time” and the “number of
combinations of n objects taken r at a time,”
and these two extremely important families
of numbers are denoted nPr and nCr,
respectively. (Some calculators use
variations of this notation, such as Pn,r and
Cn,r respectively.)
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Excursions in Modern Mathematics, 7e: 15.1 - 41
Factorial symbol
• For any integer n ≥ 0, the factorial symbol n!
is defined as follows:
• 0! = 1
• 1! = 1
• n! = n(n - 1)(n - 2) · · · 3 · 2 · 1
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Excursions in Modern Mathematics, 7e: 15.1 - 42
Example
Find each of the following
1. 4!
2. 7!
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Excursions in Modern Mathematics, 7e: 15.1 - 43
Example
ANSWER
1. 4! 4 3 2 1 24
2. 7! 7 6 5 4 3 2 1 5040
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Excursions in Modern Mathematics, 7e: 15.1 - 44
Permutation versus Combination
Summary of essential facts about the
numbers nPr and nCr,.
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Excursions in Modern Mathematics, 7e: 15.1 - 45
Example
Evaluate each:
P and 7 C4
8 3
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Excursions in Modern Mathematics, 7e: 15.1 - 46
Example
ANSWER:
8
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P3
8!
(8 3)!
8!
5!
8 7 6 5 4 3 21
5 4 3 21
8 7 6
336
Excursions in Modern Mathematics, 7e: 15.1 - 47
Permutations on the Calculator
8
MATH
To get:
8
PRB
2:nPr
3
P3
Then Enter gives 336
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Excursions in Modern Mathematics, 7e: 15.1 - 48
Example
ANSWER:
7
C4
7!
4! (7 4)!
7!
4! 3!
7 6 5 4 3 21
( 4 3 2 1) (3 2 1)
7 6 5
3 21
7 5
35
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Excursions in Modern Mathematics, 7e: 15.1 - 49
Combinations on the Calculator
7
MATH
To get:
7
PRB
3:nCr
4
C4
Then Enter gives 35
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Excursions in Modern Mathematics, 7e: 15.1 - 50
Example 15.13 Five-Card Poker Hands
We will compare two types of games: fivecard stud poker and five-card draw poker. In
both of these games a player ends up with
five cards, but there is an important difference
when analyzing the mathematics behind the
games: In five-card draw the order in which
the cards come up is irrelevant; in five-card
stud the order in which the cards come up is
extremely relevant.
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Excursions in Modern Mathematics, 7e: 15.1 - 51
Example 15.13 Five-Card Poker Hands
The reason for this is that in five-card draw all
cards are dealt down, but in five-card stud
only the first card is dealt down–the remaining
four cards are dealt up, one at a time. This
means that players can assess the relative
strengths of the other players’ hands as the
game progresses and play their hands
accordingly.
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Excursions in Modern Mathematics, 7e: 15.1 - 52
Example 15.13 Five-Card Poker Hands
Counting the number of five-card stud poker
hands is a direct application of the
multiplication rule: 52 possibilities for the first
card, 51 for the second card, 50 for the third
card, 49 for the fourth card, and 48 for the fifth
card, for an awesome total of
52 51 50 49 48 = 311,875,200
possible hands.
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Excursions in Modern Mathematics, 7e: 15.1 - 53
Example 15.13 Five-Card Poker Hands
Counting the number of five-card draw poker
hands requires a little more finesse. Here a
player gets five down cards and the hand is
the same regardless of the order in which the
cards are dealt. There are 5! = 120 different
ways in which the same set of five cards can
be ordered, so that one draw hand
corresponds to 120 different stud hands.
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Excursions in Modern Mathematics, 7e: 15.1 - 54
Example 15.13 Five-Card Poker Hands
Thus, the stud hands count is exactly 120
times bigger than the draw hands count.
Great! All we have to do then is divide the
311,875,200 (number of stud hands) by 120
and get our answer: There are 2,598,960
possible five-card draw hands.
As before, it’s more telling to look at this
answer in the uncalculated form
(52 51 50 49 48)/5! = 2,598,960.
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Excursions in Modern Mathematics, 7e: 15.1 - 55
Example 15.14 The Florida Lotto
Like many other state lotteries, the Florida
Lotto is a game in which for a small
investment of just one dollar a player has a
chance of winning tens of millions of dollars.
Enormous upside, hardly any downside–
that’s why people love playing the lottery and,
like they say, “everybody has to have a
dream.” But, in general, lotteries are a very
bad investment, even if it’s only a dollar, and
the dreams can turn to nightmares.
Why so?
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Excursions in Modern Mathematics, 7e: 15.1 - 56
Example 15.14 The Florida Lotto
In a Florida Lotto ticket, one gets to select six
numbers from 1 through 53. To win the
jackpot (there are other lesser prizes we
won’t discuss here), those six numbers have
to match the winning numbers drawn by the
lottery in any order. Since a lottery draw is
just an unordered selection of six objects (the
winning numbers) out of 53 objects (the
numbers 1 through 53), the number of
possible draws is 53C6 .
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Excursions in Modern Mathematics, 7e: 15.1 - 57
Example 15.14 The Florida Lotto
Doesn’t sound too bad until we do the
computation (or use a calculator) and realize
that
53
C6
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53 52 51 50 49 48
6!
22,957,480
Excursions in Modern Mathematics, 7e: 15.1 - 58
Example
A safe combination consists of four numbers
between 0 and 99. If four numbers are
randomly selected, determine the number of
possible combinations. Assume you are not
allowed to repeat numbers so that a
combination such as:
1-2-1-0
is not allowed. Verify that this is a permutation.
Identify both r and n.
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Excursions in Modern Mathematics, 7e: 15.1 - 59
Example
An arrangement of numbers, such that
• 4 numbers are chosen at a time from 100
distinct numbers.
• repetition of numbers is not allowed (each
number is distinct)
• the order of the numbers is important.
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Excursions in Modern Mathematics, 7e: 15.1 - 60
Examples
Solution:
100
P4
100 99 98 97 94,109,400
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Excursions in Modern Mathematics, 7e: 15.1 - 61
Examples
• Page 579, problem 34
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Excursions in Modern Mathematics, 7e: 15.1 - 62
Examples
34(a)
10
P3
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720
Excursions in Modern Mathematics, 7e: 15.1 - 63
Examples
34(b)
10
C7
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120
Excursions in Modern Mathematics, 7e: 15.1 - 64