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Chapter 8: Further Topics in Algebra

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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

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”.

SolutionThe 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)

– P(ace and 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 ofr 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