Counting Methods and Probability

Counting Methods and Probability

Chapter10.1-10.3

Determine how many different possibilities are possible:

1. There are 3 different ice cream flavors and 5 different toppings. You can have one type of ice cream and one topping.

2. You have 30 different shirts, 8 types of pants, and 4 different types of shoes. How many different ways can you dress yourself?

3. You have just enough money to go out to eat and see a movie. There are 5 different restaurants near the movie theater and 10 different movies playing.

10.1 Counting Principles and Permutations

If three events occur in m, n, and p ways, then the number of ways that all three events can occur is m x n x p.

It can be ANY number of events.

Fundamental Counting Principle

a. repetition is allowed b. repetition is not allowed. 1. A 4-digit lock with numbers 0-9. 2. A 6-digit lottery with numbers from 1-30.

3. A license plate with 3 letters followed by

4 numbers.

Determine how many different possibilities are possible if:

How many ways can you pick r things out of n, where ORDER MATTERS.

nPr=

You can plug these into your calculator. MATHPRBnPr.

Permutations

First, use the Fundamental Counting Principle. Then, use the Permutations Formula by hand. 1. A TV news director has 8 news stories to

present on the evening news. a. How many different ways can the stories be

presented? b. If only 3 stories will be presented, how many

possible ways can a lead story, a second story, and a closing story be presented?

Find the number of Permutations:

First, use the Fundamental Counting Principle. Then, use the Permutations Formula by hand. 2. 10 students at Norwin are running for

President. a. How many different ways can the students

give their speeches to the school? b. First place becomes President, second place

becomes Vice-President, third place becomes Treasurer, and fourth place becomes Secretary. How many ways can the students be P, VP, T, and S?

Find the number of Permutations:

How many different permutations can you make with the following letters:

1. ABCD 2. ABCC 3. ABBB

Permutations with Repetition:

different permutations of n objects where one object is repeated s1 times, another repeated s2 times, and so on is:

Permutations with Repetition:

1. KAYAK 2. TALLAHASSEE 3. CINCINNATI

Find the number of different permutations of the letters in:

10.1 #11-16, 32-53x3, 64-66

Homework

Place 44, 50, and 64-66 on the board. Show your work!

At your seats, answer the following questions:

1. How many different ways can Ms. Rothrauff call on students to write the above answers on the board?

2. How many different ways can you pick 4 lunch sides given that there are 10 options?

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

10.2 Combinations

How many different permutations can you make with the following letters:

1. ABCD 2. ABCC 3. ABBB How does this help prove this is true?

Permutations with Repetition:

The definition of a factorial is n!=n x (n-1)! Use this information to prove that 0!=1.

0!=1?

Discuss in your groups what you think the formula will be for Combinations (where order DOES NOT MATTER).

Consider the following: ◦ Permutation Formula from yesterday.◦ Different Permutations of ABCD picking all 4

letters.◦ If order DID NOT MATTER, how many different

possibilities would there be to order ABCD using all 4 letters?

Finding the Formula for Combinations

How many ways can you pick r things out of n, where order DOES NOT MATTER.

nCr=

You can plug these into your calculator. MATHPRBnCr.

Combinations

When finding the number of ways both event A AND event B can occur, you need to multiply.

When finding the number of ways that event A OR event B can occur, you add instead.

Pg 691

Multiple Events

Counting problems that involve phrases like “at least” or “at most” are sometimes easier to solve by subtracting possibilities you do not want from the total number of possibilities.

Pg 691

Subtracting Possibilities

The Norwin Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshman.

a. How many different committees of exactly 2 seniors and 2 juniors can be chosen?

b. How many different committees of at most 4 students can be chosen?

Multiple Events Example:

You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?

Subtracting Possibilities Example:

In a standard deck of 52 cards: 1. How many ways can you get a flush in

hearts? 2. How many ways can you get all red

cards?

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

*flush=all same suit (hearts, diamonds, etc.)

Finding card combinations:

3. How many ways can you get at most one heart?

4. How many ways can you get at least one 6?

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

Finding card combinations (cont):

10.2 #3-10, 13-18

Homework

Place numbers 14, 16, and 18 on the board. Show your work!

At your seats, answer the following question:

How many ways can you get a full house with a standard deck of 52 cards?

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

*full house=3 of the same type and 2 of the same type (QQQKK, 444JJ, 33399, etc.)

10.2 Binomial Theorem

Refer to page 692 in your books. If you arrange the values of nCr in triangular

pattern in which each row corresponds to a value of n, you get Pascal’s Triangle.

The r corresponds to the number in that row.

*You start counting with 0. Both the rows and the number in that row.*

*0C0 = 1 and is the 0th row.*

Pascal’s Triangle

1. From a collection of 7 baseball caps, you want to trade 3. Use Pascal’s Triangle to find the number of combinations of 3 caps that can be traded.

2. The 7 members of the math club chose 2 members to represent them at a meeting. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives.

Use Pascal’s Triangle:

Refer to page 693 in your books. Steps to use the Binomial Theorem: 1. Identify a, b, and n. 2. Make a list of all the C terms vertically. n=n for all C

terms, while r starts at 0 at the top and goes to n on the bottom. (There will be n+1 C terms.)

3. Next to each C term, write the a term in parenthesis. Raise each a term starting at the top to the nth power down to the bottom ending with 0th power.

4. Next to the a term, write the b term in parenthesis. Raise each b term starting at the top to the 0th power down to the bottom ending with the nth power.

5. Multiply all of the terms out and put a “+” between each new term.

Binomial Theorem

1. (x+y)6

2. (5-2y)3

3. (3x-2)4

BT Examples:

Find the coefficient of xr in the expansion of (a+b)n.

Formula: nCrarbn-r

Find a Coefficient in an Expansion:

1. Find the coefficient of x5 in the expansion of (x-3)7.

2. Find the coefficient of y3 in the expansion of (5+2y)8.

3. Find the coefficient of x3y4 in the expansion of (2x-y)7.

Coefficient Examples:

10.2 #19-33odd, 38-39, 48-49

Homework

Place 25 and 31 on the board. Show your work!!!

At your seats, try 24 and 26 on page 695.

BT Review

1. How many different possibilities are there to win a lottery if 3 numbers are drawn from 1-15…

a. With repetition? b. Without repetition? 2. What would be the probability of winning

the lottery… a. With repetition? b. Without repetition?

10.3 Probability and Odds

Theoretical Probability of event A:

P(A)=

Experimental Probability of event A:

P(A)=

Probability

You pick a card from a standard deck of 52 cards. Find the following probabilities:

1. Picking an heart. 2. Picking a red King. 3. Picking anything but an Ace. 4. Picking a number card (2-9). 5. Picking a Joker.

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

Card Probabilities:

You have an equally likely chance of picking any integer from 1-20. Find the probabilities:

1. Picking a perfect square. 2. Picking a factor of 30. 3. Picking a multiple of 3.

Number Probabilities:

Odds in favor of event A=

Odds against event A=

Odds

You pick a card from a standard deck of 52 cards. Find the following odds:

1. Odds in favor of drawing a 5. 2. Odds against drawing a diamond. 3. Odds in favor of drawing a heart. 4. Odds against drawing a Queen.

Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).

Card Odds:

You throw a dart at the board. Your dart is equally likely to hit any point inside of the board. What is the probability of getting 0 points? What is the probability of getting 50 points? Are you more likely to get 0 points or 50 points?

Geometric Probability

3in 3in 3in

0pts

50pts

10.3 #4-18even, 20-23, 35-39

Homework