Download - Chapter 8: Further Topics in Algebra
Copyright © 2007 Pearson Education, Inc. Slide 8-1
Copyright © 2007 Pearson Education, Inc. Slide 8-2
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
Copyright © 2007 Pearson Education, Inc. Slide 8-3
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}.
Copyright © 2007 Pearson Education, Inc. Slide 8-4
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
Copyright © 2007 Pearson Education, Inc. Slide 8-5
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
Copyright © 2007 Pearson Education, Inc. Slide 8-6
8.7 Finding Probabilities of Events
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
Copyright © 2007 Pearson Education, Inc. Slide 8-7
8.7 Finding Probabilities of Events
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
Copyright © 2007 Pearson Education, Inc. Slide 8-8
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.
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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
Copyright © 2007 Pearson Education, Inc. Slide 8-10
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´.
Copyright © 2007 Pearson Education, Inc. Slide 8-11
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
Copyright © 2007 Pearson Education, Inc. Slide 8-12
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
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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
Copyright © 2007 Pearson Education, Inc. Slide 8-14
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
Copyright © 2007 Pearson Education, Inc. Slide 8-15
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
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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.
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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
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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
Copyright © 2007 Pearson Education, Inc. Slide 8-19
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.
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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
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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