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# Chapter 8: Further Topics in Algebra - PowerPoint PPT Presentation

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

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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

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}.

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,

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

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

Solution

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

(b) E6 = Ø so

• 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.

• 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}.

• 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´.

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

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

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

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”.

Solution The odds in favor of a blue shirt are

or 3 to 2.

Probability of the Union of Two Events

For any events E and F,

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.

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

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

Properties of Probability

1.

2. P(a certain event) = 1;

3. P(an impossible event) = 0;

4.

5.

An experiment that consists of

• repeated independent trials,

• only two outcomes, success and failure, in

each trial,

is called a binomial experiment.

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

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