College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson
College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson
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Transcript of College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson
![Page 1: College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson](https://reader037.fdocuments.in/reader037/viewer/2022110100/56812a47550346895d8d8802/html5/thumbnails/1.jpg)
College AlgebraFifth EditionJames Stewart Lothar Redlin Saleem Watson
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Counting
and Probability10
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Probability10.3
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Overview
In the preceding chapters, we
modeled real-world situations.
• These were modeled using precise rules, such equations or functions.
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Overview
However, many of our everyday
activities are not governed by
precise rules.
• Rather, they involve randomness and uncertainty.
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Overview
How can we model such situations?
• Also, how can we find reliablepatterns in random events?
• In this section, we will see howthe ideas of probability provideanswers to these questions.
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Rolling a Die
Let’s look at a simple example.
• We roll a die, and we’re hoping to geta “two”.
• Of course, it’s impossible to predict whatnumber will show up.
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Rolling a Die
But, here’s the key idea:
• We roll the die many many times.
• Then, the number two will show upabout one-sixth of the time.
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Rolling a Die
This is because each of the six numbers is equally likely to show up.
• So, the “two” will show up abouta sixth of the time.
• If you try this experiment, you willsee that it actually works!
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Rolling a Die
We say that the probability
(or chance) of getting “two”
is 1/6.
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Picking a Card
If we pick a card from a 52-card deck,
what are the chances that it is an ace?
• Again, each card is equally likely to be picked.
• Since there are four aces, the probability (or chances) of picking an ace is 4/52.
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Probability and Science
Probability plays a key role in many sciences.
• A remarkable example of the use of probability is Gregor Mendel’s discovery of genes.
• He could not see the genes.
• His discovery was due to applying probabilistic reasoning to the patterns he saw in inherited traits.
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Probability
Today, probability is an indispensable tool for decision making.
• It is used in business, industry, government, and scientific research.
• For example, probability is used to–Determine the effectiveness of new medicine–Assess fair prices for insurance policies–Gauge public opinion on a topic
(without interviewing everyone)
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Probability
In the remaining sections of this
chapter, we will see how some of
these applications are possible.
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What is Probability?
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Terminology
To discuss probability, let’s begin by defining some terms.
• An experiment is a process, such as tossing a coin or rolling a die.
• The experiment gives definite results called the outcomes of the experiment.
– For tossing a coin, the possible outcomes are “heads” and “tails”
– For rolling a die, the outcomes are 1, 2, 3, 4, 5, and 6.
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Terminology
The sample space of an experiment is the
set of all possible outcomes.
• If we let H stand for heads and T for tails,then the sample space of the coin-tossingexperiment is S = {H, T}.
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Sample Space
The table lists some experiments and the
corresponding sample spaces.
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Experiments with Equally Likely Outcomes
We will be concerned only with experiments
for which all the outcomes are equally likely.
• We already have an intuitive feeling for whatthis means.
• When we toss a perfectly balanced coin,heads and tails are equally likely outcomes.
• This is in the sense, that if this experiment is repeated many times, we expect that aboutas many heads as tails will show up.
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Experiments and Outcomes
In any given experiment, we are often
concerned with a particular set of outcomes.
• We might be interested in a die showing an even number.
• Or, we might be interested in picking an acefrom a deck of cards.
• Any particular set of outcomes is a subset of the sample space.
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An Event—Definition
This leads to the following definition.
• If S is the sample space of an experiment,then an event is any subset of the samplespace.
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E.g. 1—Events in a Sample Space
An experiment consists of tossing a coin
three times and recording the results in order.
• The sample space is
S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
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E.g. 1—Events in a Sample Space
The event E of showing “exactly two heads” is
the subset of S.
• E consists of all outcomes with two heads.
• Thus, E = {HHT, HTH, THH}
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E.g. 1—Events in a Sample Space
The event F of showing “at least two heads”
is
F = {HHH, HHT, HTH, THH}
And, the event of showing “no heads” is
G = {TTT}
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Intuitive Notion of Probability
We are now ready to define the notion of
probability.
• Intuitively, we know that rolling a die may result in any of six equally likely outcomes.
• So, the chance of any particular outcome occurring is 1/6.
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Intuitive Notion of Probability
What is the chance of showing an even
number?
• Of the six equally likely outcomes possible, three are even numbers.
• So it is reasonable to say that the chance of showing an even number is 3/6 = 1/2.
• This reasoning is the intuitive basis for the following definition of probability.
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Probability—Definition
Let S be the sample space of an experiment
in which all outcomes are equally likely.
• Let E be an event.
• The probability of E, written P(E), is
( ) number of elements in
( )( ) number of elements in
n E EP E
n S S
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Values of a Probability
Notice that 0 ≤ n(E) ≤ n(S).
• So, the probability P(E) of an event is a number between 0 and 1.
• That is, 0 ≤ P(E) ≤ 1.
• The closer the probability is to 1, the more likely the event is to happen.
• The close to 0, the less likely.
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Values of a Probability
If P(E) = 1, then E is called the certain event.
• If P(E) = 0, then E is called the impossible event.
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E.g. 2—Finding the Probability of an Event
A coin is tossed three times, and the results
are recorded.
• What is the probability of getting exactlytwo heads?
• At least two heads?
• No heads?
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E.g. 2—Finding the Probability of an Event
By the results of Example 1, the sample space S of this experiment contains eight outcomes.
• The event E of getting “exactly two heads”contains three outcomes.
• They are {HHT, HTH, THH}.
• So, by the definition of probability,
( ) 3
( )( ) 8
n EP E
n S
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E.g. 2—Finding the Probability of an Event
Similarly, the event F of getting “at least two
heads” has four outcomes.
• They are {HHH, HHT, HTH, THH}.
• So,
( ) 4 1
( )( ) 8 2
n EP E
n S
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E.g. 2—Finding the Probability of an Event
The event G of getting “no heads” has one
element, so
( ) 1
( )( ) 8
n EP E
n S
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Calculating Probability
by Counting
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Calculating Probability by Counting
To find the probability of an event:
• We do not need to list all the elements in the sample space and the event.
• What we do need is the number of elementsin these sets.
• The counting techniques that we learned inthe preceding sections will be very usefulhere.
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E.g. 3—Finding the Probability of an Event
A five-card poker hand is drawn from a
standard 52-card deck.
• What is the probability that all five cardsare spades?
• The experiment here consists of choosingfive cards from the deck.
• The sample space S consists of all possiblefive-card hands.
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E.g. 3—Finding the Probability of an Event
Thus, the number of elements in the sample
space is
( ) (52,5)
52!
5!(52 5)!
2,598,960
n S C
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E.g. 3—Finding the Probability of an Event
The event E that we are interested in consists
of choosing five spades.
• Since the deck contains only 13 spades,the number of ways of choosing five spadesis
( ) (13,5)
13!
5!(13 5)!
1,287
n E C
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E.g. 3—Finding the Probability of an Event
Thus, the probability of drawing five spades is
( )( )
( )
1,287
2,598,960
0.0005
n EP E
N S
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Understanding a Probability
What does the answer to Example 3 tell us?
• Since 0.0005 = 1/2000, this means that if youplay poker many, many times, on averageyou will be dealt a hand consisting of onlyspades about once every 2000 hands.
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E.g. 4—Finding the Probability of an Event
A bag contains 20 tennis balls.
• Four of the balls are defective.
• If two balls are selected at random fromthe bag, what is the probability that bothare defective?
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E.g. 4—Finding the Probability of an Event
The experiment consists of choosing two
balls from 20.
• So, the number of elements in the sample space S is C(20, 2).
• Since there are four defective balls,the number of ways of picking two defective balls is C(4, 2).
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E.g. 4—Finding the Probability of an Event
Thus, the probability of the event E of picking
two defective balls is
( )( )
( )
(4,2)
(20,2)
6
200.032
n EP E
n S
C
C
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Complement of an Event
The complement of an event E is the set
of outcomes in the sample space that is
not in E.
• We denote the complement of an event Eby E′.
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Complement of an Event
We can calculate the probability of E′ using
the definition and the fact that
n(E′) = n(S) – n(E)
• So, we have
( ') ( ) ( ) ( ) ( )( ')
( ) ( ) ( ) ( )
1 ( )
n E n S n E n S n EP E
n S n S n S n S
P E
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Probability of the Complement of an Event
Let S be the sample space of an experiment,
and E and event.
• Then
( ') 1 ( )P E P E
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Probability of the Complement of an Event
This is an extremely useful result.
• It is often difficult to calculate the probabilityof an event E.
• But, it is easy to find the probability of E′.
• From this, P(E) can be calculated immediatelyby using this formula.
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E.g. 5—The Probability of the Complement of an Event
An urn contains 10 red balls and 15 blue
balls.
• Six balls are drawn at random from the urn.
• What is the probability that at least one ballis red?
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E.g. 5—The Probability of the Complement of an Event
Let E be the event that at least one red ball is
drawn.
• It is tedious to count all the possible waysin which one or more of the balls drawnare red.
• So let’s consider E′, the complement of thisevent.
• E′ is the event that none of the balls drawnare red.
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E.g. 5—The Probability of the Complement of an Event
The number of ways of choosing 6 blue balls
from the 15 balls is C(15, 6).
• The number of ways of choosing 6 ballsfrom the 25 ball is C(25, 6).
• Thus,
( ') (15,6) 5,005( ')
( ) (25,6) 177,100
13
460
n E CP E
n S C
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E.g. 5—The Probability of the Complement of an Event
By the formula for the complement of an
event, we have
( ) 1 ( ')
131
460447
4600.97
P E P E
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The Union of Events
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The Union of Events
If E and F are events, what is the probability
that E or F occurs?
• The word or indicates that we want the probability of the union of these events.
• That is, .E F
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The Union of Events
So, we need to find the number of elements
in .
• If we simply add the number of element in E to the number of elements in F, then we wouldbe counting the elements in the overlap twice.
• Once in E and once in F.
E F
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The Union of Events
So to get the correct total, we must subtract
the number of elements in .
• Thus,
E F
( ) ( ) ( ) ( )n E F n E n F n E F
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The Union of Events
Using the formula for probability,
we get
• We have just proved the following.
( ) ( ) ( ) ( )( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
n E F n E n F n E FP E F
n S n S
n E n F n E F
n S n S n S
P E P F P E F
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The Probability of The Union of Events
If E and F are events in a sample space S,
then the probability of E or F is
( ) ( ) ( ) ( )P E F P E P F P E F
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E.g. 6—The Probability of the Union of an Event
What is the probability that a card drawn at
random from a standard 52-card deck is
either a face card or a spade?
• We let E and F denote the following events:
E: The card is a face card.
F: The card is a spade.
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E.g. 6—The Probability of the Union of an Event
There are 12 face cards and 13 spades in a
51-card deck, so
12 13
( ) and ( )52 52
P E P F
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E.g. 6—The Probability of the Union of an Event
Since three cards are simultaneously face
cards and spades, we have
3
( )52
P E F
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E.g. 6—The Probability of the Union of an Event
Thus, by the formula for the probability of the
union of two events, we have
( ) ( ) ( ) ( )
12 13 3
52 52 5211
26
P E F P E P F P E F
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The Union of
Mutually Exclusive Events
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Mutually Exclusive Events
Two events that have no outcome in common
are said to be mutually exclusive.
• This is illustrated in the figure.
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A Mutually Exclusive Event
For example, draw a card from a deck.
• Consider the eventsE: The card is an ace.F: The card is a queen.
• They are mutually exclusive because a cardcannot be both an ace and a queen.
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Mutually Exclusive Events
If E and F are mutually exclusive events,
then contains no elements.
• Thus,
• So,
• We have proved the following formula.
( ) 0P E F
( ) ( ) ( ) ( )
( ) ( )
P E F P E P F P E F
P E P F
E F
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Probability of the Union of Mutually Exclusive Events
If E and F are mutually exclusive events
in a sample space S, then the probability
of E or F is
( ) ( ) ( )P E F P E P F
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Multiple Mutually Exclusive Events
There is a natural extension of this formula
for any number of mutually exclusive events:
• If E1, E2, … , En are pairwise mutually exclusive, then
1 2 1 2... ...n nP E E E P E P E P E
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E.g. 7—The Union of Mutually Exclusive Events
A card is drawn at random from a standard
deck of 52 cards.
• What is the probability that the card is eithera seven or a face card?
• Let E and F denote the following events:E: The card is a seven.F: The card is a face card.
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E.g. 7—The Union of Mutually Exclusive Events
A card cannot be both a seven and
a face card.
• Thus, the events are mutually exclusive.
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E.g. 7—The Union of Mutually Exclusive Events
We want the probability of E or F.
• In other words, the probability of . E F
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E.g. 7—The Union of Mutually Exclusive Events
By the formula,
( ) ( ) ( )
4 12
52 524
13
P E F P E P F
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The Intersections of
Independent Events
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The Intersection of Events
We have considered the probability of events
joined by the word or.
• That is, the union of events.
• Now, we study the probability of events joinedby the word and.
– In other words, the intersection of events.
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The Intersection of Independent Events
When the occurrence of one event does not affect the probability of another event:
• We say that the events are independent.
• For instance, if a balanced coin is tossed,the probability of showing heads on thesecond toss is 1/2.
– This is regardless of the outcome of the firsttoss.
– So, any two tosses of a coin are independent.
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Probability of the Intersection of Independent Events
If E and F are independent events in a
sample space S, then the probability
of E and F is
( ) ( ) ( )P E F P E P F
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E.g. 8—The Probability of Independent Events
A jar contains five red balls and four black
balls.
• A ball is drawn at random from the jar and then replaced.
• Then, another ball is picked.
• What is the probability that both balls are red?
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E.g. 8—The Probability of Independent Events
The events are independent.
• The probability that the first ball is red is 5/9.
• The probability that the second ball is red isalso 5/9.
• Thus, the probability that both balls are red is
5 5 25
9 9 810.31
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E.g. 9—The Birthday Problem
What is the probability that in a class of
35 students, at least two have the same
birthdays?
• It is reasonable to assume that the 35 birthdaysare independent.
• It can also be assumed that each day of the 365 days in a year is equally likely as a dateof birth.
– We ignore February 29.
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E.g. 9—The Birthday Problem
Let E be the event that two of the students
have the same birthday.
• It is tedious to list all the possible ways in whichat least two of the students have matching birthdays.
• So, we consider the complementary event E′.
• That is, that no two students have the samebirthday.
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E.g. 9—The Birthday Problem
To find this probability, we consider the students one at a time.
• The probability that the first student hasa birthday is 1.
• The probability that the second has a birthdaydifferent from the first is 364/365.
• The probability that the third has a birthdaydifferent from the first two is 363/365.
• And so on.
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E.g. 9—The Birthday Problem
Thus,
• So,
( ) 1 ( ')
1 0.186 0.814
P E P E
364 363 362 331
( ') 1 ...365 365 365 365
0.186
P E
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The Birthday Paradox
Most people are surprised that the probability
in Example 9 is so high.
• For this reason, this problem is sometimescalled the “birthday paradox”.
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The Birthday Paradox
The table below gives the probability that
two people in a group
will share the same
birthday for groups
of various sizes.