Math 2 Guided Notes(Chapter 6)

Lesson 6.1 Exploring Counting Problems

Key terms: Tree diagram –

Outcome –

Event –

Example:

A summer camp offers a choice of swimming, tennis, or softball in the morning and gymnastics or crafts in the afternoon. How many different pairings of one morning and one afternoon activity can be scheduled? Make a tree diagram.

Example:

A die is rolled and a penny is tossed. How many outcomes are there? Make a list and/or a tree diagram.

Lesson 6-1 Assignment: (p.299) 1-4, 7-9, 12, 15-19, 21-30Extra Practice: (p.623) 1-3

Math 2 Guided Notes(Chapter 6)

Lesson 6.2 Counting and Permutations

Key terms: Factorial –

Permutation –

I. Multiplication counting principle

This is used to determine the number of possible outcomes for an event found by multiplying the number of choices at each stage

Example:

A high school assigns each student a 4-digit ID number using the digits 0 through 9. Digits may be repeated, but the first digit cannot be 0. How many ID numbers are possible?

Example:

How many 3-letter arrangements of the letters in the word PART are possible if no letter canbe repeated?

II. Permutations of n items arranged r at a time

o the number of n items arranged r at a time (r<n)

Formula: nPr =

Example:

Find the value of: 7P4

Example:

Find the value of: 5P3

Example:

Awards are given for the first five finishers. There are 12 people entered in the race. How many different ways could the awards be won?

Example:

How many different 3 letter arrangements can be made from the word BABY?

Example:

How many different arrangements can be made from the word FOOTBALL?

Example:

How many different arrangements can be made from the word MISSISSIPPI?

Lesson 6-2 Assignment: (p.306) 1-10,13a, 14a, 16-28, 32-39Extra Practice: (p.623) 4,5

Math 2 Guided Notes(Chapter 6)

Lesson 6.3 Probability and Odds

Key terms: Mutually Exclusive Events –

Complementary Events –

Odds in Favor –

Odds Against –

I. Probability

Examples:

A fair die is rolled

1. P(5) 2. P(multiple of 3) 3. P(5 or a multiple of 3)

Example:

In a card game, you are dealing the first card from a well-mixed standard deck. Find the probability of each event.

a) Dealing an ace or a jack

b) Dealing a face card or a spade

II. Complementary Events

The sum of the probabilities of complementary events is 1

P(E) = 1 – P(not E) and P(not E) = 1 – P(E)

Example:

The numbers 1 through 100 are written on cards. One card is drawn. Find the P(not a perfect square).

III. Odds

Example:

There are 1500 tickets sold for a raffle. There is one grand prize of a stereo system and 5 cash prizes.

1. Find the odds in favor of winning a cash prize.

2. Find the odds against winning any prize.

Example:

Find the probability that no two people in a group of five people have the same birthday (month and day).

Lesson 6-3 Assignment: (p.315) 1-13, 15-32, 34-39, 41-46Extra Practice:(p.623) 6, 7Worksheet: Conditional Probability

Math 2 Guided Notes(Chapter 6)

Lesson 6.4 Compound Events

Key terms: Compound Events –

Independent Events –

Dependent Events –

Sample Space –

I. Probability and Independent Events

(Multiply the probabilities together)

Example:

A bag contains 3 red marbles and 5 blue marbles. One marble is chosen at random and thenreplaced. A second marble is drawn. Find the P(red, then blue).

Example:

Two draws are made from a standard deck of 52 cards. There is replacement after the first draw.Find the P(queen, face card).

II. Dependent Events

Example:

A bag has 3 red marbles and 5 blue marbles. One marble is chosen and not replaced. A second marble is drawn. Find the P(red, red).

Example:

Two draws are made from a standard deck of 52 cards. The card is not replaced after the first draw. Find the P(two diamonds).

Lesson 6-4 Assignment: (p.323) 1-5, 10-12, 16-19, 25-29Checkpoint: (p.327) 1-9Worksheet: Area Probability

Math 2 Guided Notes(Chapter 6)

Lesson 6.5 Combinations

Key term: Combination –

I. Combinations of n items selected r at a time

Formula: nCr = =

Examples:

1. 5C3 2. 8C5

Example:

A team plans to choose 3 of the colors represented in the 5 rings of the Olympic flag for team uniforms. Colors are blue, yellow, black, red, and green. How many different 3-color combinationscan be chosen?

Tell whether the problem is asking for permutations or combinations, the answer the question.

Example:

How many ways can 4 posters be arranged on the wall?

Example:

How many groups of 4 posters can be chosen from 7 posters?

Lesson 6-5 Assignment: (p.330) 1-4, 6-13, 16-20, 25-30Extra Practice: (p.624) 10-15Worksheet: Expected Value

Math 2 Guided Notes(Chapter 6)

Lesson 6.6 Pascal’s Triangle

Key term: Pascal’s Triangle – a triangular array of numbers with 1’s at the beginning and end of each row and with

all other numbers being the sum of the two numbers just above.

Another way to find the entries of Pascal’s triangle uses combinations The number in row n, diagonal r of Pascal’s triangle is nCr

The sum of the numbers in row n is 2n

Examples:

Use Pascal’s triangle to find:1. 7C5 2. 9C4

3. Tom has a picture frame that holds 4 pictures. In how many ways can he choose among pictures of 8 friends to fill the frame?

Lesson 6-6 Assignment: (p.336) 1,5-12, 14-17, 24-28Extra Practice: (p.624) 16-23

Math 2 Guided Notes(Chapter 6)

Lesson 6.7 Binomial Experiments with p = ½

Key term:Binomial experiment –

I. P(E) = # of favorable outcomes / total # of possible outcomes for a binomial experiment with P(success) = ½ , this can be found using Pascal’s triangle

Example:

Three coins are tossed. Find the theoretical probability of getting exactly two heads.

Example:

Find the probability of a 5-child family having exactly 4 boys.

Example:

Find the probability of a 5-child family having at least 4 boys.

Lesson 6-7 Assignment: (p.341) 3-5, 7, 8, 11-16, 18-21Checkpoint: 1-6

Math 2 Guided Notes(Chapter 6)

Lesson 6.8 Binomial Experiments with p ≠ ½

Key term: Probability tree diagram – a tree diagram with the probability of each branch written on that branch

I. Binomial experiments with p ≠ ½

Formula: P(E) = favorable / total outcomes This can’t be used if the outcomes are not equally likely but it is still relatively straight forward

(events are mutually exclusive and the events are independent)

Example:

A trick coin is made so that when the coin is tossed, P(Heads) = .6 The coin is tossed three times. Find the probability of getting exactly two heads.

Example:

At a school, about 20% of the students walk to school. Find the probability that exactly 3 out of 4 students chosen at random walk to school.

Lesson 6-8 Assignment: (p.347) 2-10, 13, 19-23Extra Practice: (624) 28, 29

Math 2 Guided Notes(Chapter 6)

Lesson 6.9 The Binomial Theorem

Key terms: Binomial –

Factored form –

Expanded form –

I. Expanded form of a binomial use Pascal’s triangle to find the coefficients in expanded form of a power of a binomial

Example:

Write (x + y)4 in expanded form.

Step 1: 4 is the power so use row 4 of Pascal’s triangle. Write the 5 #’s leaving a space between them for the variables.

Step 2: Write the powers of x next to each coefficient from left to right, starting with 4 and decreasing by 1 for each term.

Step 3: Write the powers of y next to each power of x from left to right starting with 0 and increasing by 1 for each term.

Step 4: Simplify.

Example:

Write (k + 2)5 in expanded form.

Example:

Expand (2 + 3y)3

Example:

Expand (y – 1)3

Lesson 6-9 Assignment: (p.354) 4-20, 26-31Extra Practice: (p.624) 30-33Review: (p.359) 1-23