Chapter 8: Further Topics in Algebra

Copyright © 2007 Pearson Education, Inc. Slide 8-1


Chapter 8: Further Topics in Algebra

8.1 Sequences and Series

8.2 Arithmetic Sequences and Series

8.3 Geometric Sequences and Series

8.4 The Binomial Theorem

8.5 Mathematical Induction

8.6 Counting Theory

8.7 Probability


8.7 Probability

Basic Concepts

• An experiment has one or more outcomes. The outcome of rolling a die is a number from 1 to 6.

• The sample space is the set of all possible outcomes for an experiment. The sample space for a dice roll is {1, 2, 3, 4, 5, 6}.

• Any subset of the sample space is called an event. The event of rolling an even number with one roll of a die is {2, 4, 6}.


8.7 Probability

Probability of an Event E

In a sample space with equally likely outcomes, the probability of an event E, written P(E), is the ratio of the number of outcomes in sample space S that belong to E, n(E), to the total number of outcomes in sample space S, n(S). That is,

( )( ) .

( )

n EP E

n S


8.7 Finding Probabilities of Events

Example A single die is rolled. Give the probabilityof each event.

(a) E3 : the number showing is even

(b) E4 : the number showing is greater than 4

(c) E5 : the number showing is less than 7

(d) E6 : the number showing is 7



Solution The sample space S is {1, 2, 3, 4, 5, 6} so

n(S) = 6.

(a) E3 = {2, 4, 6} so

(b) E4= {5, 6} so

33

( ) 3 1( ) .

( ) 6 2

n EP E

n S

44

( ) 2 1( ) .

( ) 6 3

n EP E

n S



Solution

(c) E5 = {1, 2, 3, 4, 5, 6} so

(b) E6 = Ø so

55

( ) 6( ) 1 .

( ) 6

n EP E

n S

66

( ) 0( ) 0 .

( ) 6

n EP E

n S


8.7 Probability

• For an event E, P(E) is between 0 and 1 inclusive.

• An event that is certain to occur always has probability 1.

• The probability of an impossible event is always 0.


8.7 Complements and Venn Diagrams

• The set of all outcomes in a sample space that do not belong to event E is called the complement of E, written E´. If S = {1, 2, 3, 4, 5, 6} and E = {2, 4, 6} then E´ = {1, 3, 5}.

• ' , 'E E S E E


8.7 Complements and Venn Diagrams

• Probability concepts can be illustrated with Venn diagrams. The rectangle represents the sample space in an experiment. The area inside the circle represents event E; and the area inside the rectangle but outside the circle, represents event E´.


8.7 Using the Complement

Example A card is drawn from a well-shuffled

deck, find the probability of event E, the card is

an ace, and event E´.

Solution There are 4 aces in the deck of 52

cards and 48 cards that are not aces. Therefore

( ) 4 1 ( ') 48 12

( ) ( ') .( ) 52 13 ( ) 52 13

n E n EP E P E

n S n S


8.7 Odds

The odds in favor of an event E are expressed as the

ratio of P(E) to P(E´) or as the fraction

( ).

( ')

P E

P E


8.7 Finding Odds in Favor of an Event

Example A shirt is selected at random from a dark

closet containing 6 blue shirts and 4 shirts that are

not blue. Find the odds in favor of a blue shirt

being selected.

Solution E is the event “blue shirt is selected”.

6 3 4 2

( ) , ( ') .10 5 10 5

P E P E


8.7 Finding Odds in Favor of an Event

Solution The odds in favor of a blue shirt are

or 3 to 2.

33 2 35( ) to ( ') to

25 5 25

P E P E


8.7 Probability

Probability of the Union of Two Events

For any events E and F,

( or ) ( ) ( ) ( ) ( ) .P E F P E F P E P F P E F


8.7 Finding Probabilities of Unions

Example One card is drawn from a well-shuffled

deck of 52 cards. What is the probability of each

event?

(a) The card is an ace or a spade.

(b) The card is a 3 or a king.


8.7 Finding Probabilities of Unions

Solution (a) P(ace or space) = P(ace) + P(spade)

– P(ace and spade)

(b) P(3 or K) = P(3) + P(K) – P(3 and K)

4 13 1 16 4.

52 52 52 52 13

4 4 8 20 .

52 52 52 13


8.7 Probability

Properties of Probability

1.

2. P(a certain event) = 1;

3. P(an impossible event) = 0;

4.

5. ( or ) ( ) ( ) ( ) ( ) .P E F P E F P E P F P E F

0 ( ) 1;P E

( ') 1 ( );P E P E


8.7 Binomial Probability

An experiment that consists of

• repeated independent trials,• only two outcomes, success and

failure, in each trial,

is called a binomial experiment.


8.7 Binomial Probability

Let the probability of success in one trial be p.

Then the probability of failure is 1 – p.

The probability of r successes in n trials is given by

(1 ) .r n rnp p

r


8.7 Finding Binomial Probabilities

Example An experiment consists of rolling a die 10

times. Find the probability that exactly 4 tosses result

in a 3.

Solution Here , n = 10 and r = 4. The required probability is

4 10 4 4 610 1 1 1 51 210 .054 .

4 6 6 6 6

1

6p

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Chapter 8: Further Topics in Algebra