Chapter 8 further topics in algebra
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Chapter 8: Further Topics in Algebra. 8.1Sequences and Series 8.2Arithmetic Sequences and Series 8.3Geometric Sequences and Series 8.4The Binomial Theorem 8.5Mathematical Induction 8.6Counting Theory 8.7Probability. 8.7 Probability. Basic Concepts

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Chapter 8: Further Topics in Algebra

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Chapter 8 further topics in algebra

Chapter 8: Further Topics in Algebra

8.1Sequences and Series

8.2Arithmetic Sequences and Series

8.3Geometric Sequences and Series

8.4The Binomial Theorem

8.5Mathematical Induction

8.6Counting Theory

8.7Probability


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 probability1

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,


8 7 finding probabilities of events

8.7 Finding Probabilities of Events

Example A single die is rolled. Give the probability

of 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


8 7 finding probabilities of events1

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


8 7 finding probabilities of events2

8.7 Finding Probabilities of Events

Solution

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

(b) E6 = Ø so


8 7 probability2

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

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}.


8 7 complements and venn diagrams1

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

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


8 7 odds

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


8 7 finding odds in favor of an event

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.

SolutionE is the event “blue shirt is selected”.


8 7 finding odds in favor of an event1

8.7 Finding Odds in Favor of an Event

SolutionThe odds in favor of a blue shirt are

or 3 to 2.


8 7 probability3

8.7 Probability

Probability of the Union of Two Events

For any events E and F,


8 7 finding probabilities of unions

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 unions1

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)


8 7 probability4

8.7 Probability

Properties of Probability

1.

2. P(a certain event) = 1;

3. P(an impossible event) = 0;

4.

5.


8 7 binomial probability

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 probability1

8.7 Binomial Probability

Let the probability of success in one trial be p.

Then the probability of failure is 1 – p.

The probability ofr successes in n trials is given by


8 7 finding binomial probabilities

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


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