CHAPTER 11: FURTHER

TOPICS IN ALGEBRA

March 10, 2009

TOPICS Counting

Fundamental Counting principleFactorialsPermutations and combinations

ProbabilityComplementary eventsCompound events Independent eventsDependent events

FUNDAMENTAL COUNTING PRINCIPLE Consider this: You have an exam with five true or false

questions. How many different ways are there to answer the test?

The fundamental counting principle states that: "If there are r ways to do one thing, and s ways to do another, and t ways to do a third thing, and so on ..., then the number of ways of doing all those things at once is    r * s * t etc. ..."

CONSIDER THIS PROBLEM You have four pairs of pants, five shirts,

and six pairs of shoes. How many outfits can you make?

CONSIDER THIS PROBLEM

You are making license plates, but you are limited to letters in the first three spaces and numbers in the final three spaces. How many license plates could you make?

PERMUTATIONS A permutation is an ordering or

arrangement. Suppose that there are five students

who need to ask for help on their pre-calculus homework. How many different orders could there be?

What if there were seven students instead of five?

FACTORIALS A more efficient way to find the products

is the previous problems is factorial notation.

Factorial notation is defined as follows:For any natural number n, n! = n(n-1)(n-2)…(2)(1)0! = 1

Find 2!, 5!, and 5! Use your calculator to find 13! And 20!

PERMUTATIONS OF N ELEMENTS TAKEN R AT A TIME There are 15 students in speech. How

many arrangements are there for 3 students to give a speech?

P(n, r) represents the number of permutations for n elements, taken r at a time.

P (n, r) = n!/[(n-r)!] = n(n-1)(n-2) {for a total of r factors}

Try these:P(7, 2)P(10, 3)

COMBINATIONS A combination is not a ordering or

arrangement, but a subset of a set of elements.

Consider: you have 4 people in class ready to present, but you want to pick a team of two people to make a presentation. How many choices do you have?

C(n, r) represents this choice. C(n, r) = P(n, r)/r! = n!/[(n-r)!*r!) Try these:

C(10, 2) C (30, 25)

COMBINATIONS VS PERMUTATIONS In permutations, order matters.

AB ≠ BA In combinations, order does not matter.

AB = BA

PROBABILITY Probability is the likelihood that an

event will occur. It is between 0 and 1. A probability of 0 means an event will

never happen. A probability of 1 means an event is

certain to happen. A few terms:

Outcome: the result from an experimentSample space: the set of all outcomesEvent: subset of sample space

COMPUTING PROBABILITY Probability is computed by dividing the

favorable outcomes by possible outcomes.P(E) = n(E)/n(S)

Where n(E) is the number of outcomes that are favorable and n(S) is the total outcomes

Find the probability of drawing a queen from a deck of cards.

PROBABILITY OF A COMPLEMENT An event and its complement take up

the entire sample space. This is often shown as E and E’. For example, when rolling dice, the

probability of rolling a six and the probability of not rolling a six are complements. What is the probability of each?

P(E)+ P(E’) = 1

COMPOUND EVENTS This is the probability of two events

occurring. It is measured by the probability of each

event occurring minus any overlap. Try this one:

What is the probability of rolling two dice and getting either the sum of eight or rolling a pair?

PROBABILITY OF INDEPENDENT EVENTS Suppose you toss a coin and then toss it

again. What is the probability that you will get two heads?

This is the probability of independent events.

It is the probability of one event times the probability of the other.

PROBABILITY OF DEPENDENT EVENTS Imagine that you trying to find the

probability of drawing two hearts from a deck of cards. In scenario 1, the card you have drawn is replaced. In scenario 2, the card is not replaced. What is the probability of each scenario?

Scenario 2 is the probability of dependent events.

This is the probability of the 1st event and then the probability of the 2nd event, assuming that the 1st event happened.

YOUR ASSIGNMENT Pages 926-928 #32-44 even, #56-68

even Page 951 #22-30 even