College AlgebraFifth EditionJames Stewart Lothar Redlin Saleem Watson

Counting

and Probability10

Probability10.3

Overview

In the preceding chapters, we

modeled real-world situations.

• These were modeled using precise rules, such equations or functions.

Overview

However, many of our everyday

activities are not governed by

precise rules.

• Rather, they involve randomness and uncertainty.

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.

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.

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.

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!

Rolling a Die

We say that the probability

(or chance) of getting “two”

is 1/6.

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.

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.

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)

Probability

In the remaining sections of this

chapter, we will see how some of

these applications are possible.

What is Probability?

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.

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}.

Sample Space

The table lists some experiments and the

corresponding sample spaces.

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.

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.

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.

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}

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}

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}

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.

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.

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

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.

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.

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?

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

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

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

Calculating Probability

by Counting

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.

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.

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

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

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

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.

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?

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).

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

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′.

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

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

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.

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?

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.

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

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

The Union of Events

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

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

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

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

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

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.

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

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

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

The Union of

Mutually Exclusive Events

Mutually Exclusive Events

Two events that have no outcome in common

are said to be mutually exclusive.

• This is illustrated in the figure.

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.

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

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

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

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.

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.

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

E.g. 7—The Union of Mutually Exclusive Events

By the formula,

( ) ( ) ( )

4 12

52 524

13

P E F P E P F

The Intersections of

Independent Events

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.

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.

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

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?

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

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.

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.

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.

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

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”.

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.