4.3 Counting Techniques Prob & Stats

Tree Diagrams When calculating probabilities, you need to know the total number of _____________ in the ______________. When calculating probabilities, you need to know the total number of _____________ in the ______________. outcomes sample space

Tree Diagrams Example Use a TREE DIAGRAM to list the sample space of 2 coin flips. Use a TREE DIAGRAM to list the sample space of 2 coin flips. YOU On the first flip you could get….. H T If you got H Now you could get… If you got TNow you could get… H T H T HH HT TH TT Sample Space

Tree Diagram Example Mr. Arnold’s Closet Mr. Arnold’s Closet 3 Shirts2 Pants 2 Pairs of Shoes

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits 1 2

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits

Dress Mr. Arnold List all of Mr. Arnold’s outfits List all of Mr. Arnold’s outfits If Mr. Arnold picks an outfit with his eyes closed……. P(brown shoe) = 6/12 1/2 P(polo) = 4/12 1/3 P(lookin’ cool) = 1

Multiplication Rule of Counting The size of the sample space is the ___________ of our probability The size of the sample space is the ___________ of our probability So we don’t always need to know what each outcome is, just the of outcomes. So we don’t always need to know what each outcome is, just the of outcomes. denominator number

Multiplication Rule of Compound Events If… X = total number of outcomes for event A X = total number of outcomes for event A Y = total number of outcomes for event B Y = total number of outcomes for event B Then number of outcomes for A followed by B = _________ Then number of outcomes for A followed by B = _________ x times y

Multiplication Rule: Dress Mr. Arnold Mr. Reed had 3 EVENTS Mr. Reed had 3 EVENTS pants shoes shirts How many outcomes are there for EACH EVENT? 223 2(2)(3) = 12 OUTFITS

Permutations Sometimes we are concerned with how many ways a group of objects can be __________. Sometimes we are concerned with how many ways a group of objects can be __________. arranged How many ways to arrange books on a shelfHow many ways to arrange books on a shelf How many ways a group of people can stand in lineHow many ways a group of people can stand in line How many ways to scramble a word’s lettersHow many ways to scramble a word’s letters

Wonder Woman’s invisible plane has 3 chairs. Wonder Woman’s invisible plane has 3 chairs. There are 3 people who need a lift. There are 3 people who need a lift. How many seating options are there? How many seating options are there? Example: 3 People, 3 Chairs Superman driving Batman driving Wonder Woman driving 6 Seating Options! Think of each chair as an EVENT How many ways could the 1 st chair be filled? Now that the 1 st is filled? How many options for 2 nd ? Now the first 2 are filled. How many ways to fill 3 rd ? 3(2)(1) = 6 OPTIONS

Example: 5 People, 5 Chairs The batmobile has 5 chairs. The batmobile has 5 chairs. There are 5 people who need a lift. There are 5 people who need a lift. How many seating options are there? How many seating options are there? Multiply!! =120 Seating Options This is a PERMUTATION of 5 objects

Commercial Break: FACTORIAL denoted with !  denoted with !  Multiply all integers ≤ the number  Multiply all integers ≤ the number  0! = 0! = 1! = 1! = Calculate 6! Calculate 6! What is 6! / 5!? What is 6! / 5!? 5! 5! = 5(4)(3)(2)(1) = ! = 6(5)(4)(3)(2)(1) = 720

Commercial Break: FACTORIAL denoted with !  denoted with !  Multiply all integers ≤ the number  Multiply all integers ≤ the number  0! = 0! = 1! = 1! = Calculate 6! Calculate 6! What is 6! / 5!? What is 6! / 5!? 5! 5(4)(3)(2)(1) 1 1 6(5)(4)(3)(2)(1) =6=6

Example: 5 People, 5 Chairs The batmobile has 5 chairs. The batmobile has 5 chairs. There are 5 people who need a lift. There are 5 people who need a lift. How many seating options are there? How many seating options are there? Multiply!! =120 Seating Options This is a PERMUTATION of 5 objects 5!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them What if I choose these 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them What if I choose these 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them What if I choose these 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them BUT… What if I choose THESE 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them BUT… What if I choose THESE 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them BUT… What if I choose THESE 3? Think of the possibilities!

Permutations: Not everyone gets a seat! It’s time for annual Justice League softball game. It’s time for annual Justice League softball game. How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? How many ways could your assign people to play 1 st, 2 nd, and 3 rd base? You have to choose 3 AND arrange them BUT… What if I choose THESE 3? Think of the possibilities! This is going to take FOREVER

You have 3 EVENTS? How many outcomes for each event How many outcomes for each event You have to choose 3 AND arrange them How many outcomes for this event! 5

You have 3 EVENTS? You have to choose 3 AND arrange them How many outcomes for this event! 4 Now someone is on FIRST 5

You have 3 EVENTS? You have to choose 3 AND arrange them And on SECOND 4 Now someone is on FIRST 5 3 How many outcomes for this event! 5(4)(3) = 120 POSSIBLITIES

Permutation Formula You have You have You select You select This is the number of ways you could select and arrange in order: This is the number of ways you could select and arrange in order: Another common notation for a permutation is nPr n objects r objects

You have to choose 3 AND arrange them n = r = 5 people to choose from 3 spots to fill 5! Softball Permutation Revisited (5 – 3)! 5(4)(3)(2)(1) 2!2(1) 5(4)(3) = 120 POSSIBLITIES

Combinations Sometimes, we are only concerned with a group and in which they are selected. Sometimes, we are only concerned with a group and in which they are selected. A gives the number of ways to of r objects from a group of size n. A gives the number of ways to of r objects from a group of size n. selecting not the order combination select a sample

C ombination: Duty Calls There is an evil monster threatening the city. There is an evil monster threatening the city. The mayor calls the Justice League. The mayor calls the Justice League. He requests that 3 members be sent to combat the menace. He requests that 3 members be sent to combat the menace. The Justice League draws 3 names out of a hat to decide. The Justice League draws 3 names out of a hat to decide. Does it matter who is selected first? Does it matter who is selected first? Does it matter who is selected last? Does it matter who is selected last? NOPE

C ombination: Duty Calls Let’s look at the drawing possibilities STOP! This is a waste of time These are all the SAME: The monster doesn’t care who got drawn first. All these outcomes = same people pounding his face We’ll count them as ONE OUTCOME These are all the SAME: The monster doesn’t care who got drawn first. All these outcomes = same people pounding his face We’ll count them as ONE OUTCOME

C ombination: Duty Calls Okay, let’s consider other outcomes 10 Possible Outcomes!

Combination Formula You have You have You want a group of You want a group of You what order they are selected in You what order they are selected in n objects r objects DON’T CARE Combinations are also denoted nCr Read “n choose r”

D uty Calls: Revisited n = r = 5 people to choose from 3 spots to fill ORDER DOESN’T MATTER 5! 3!(5 - 3)! 3!(2)! 5(4)(3)(2)(1) 3(2)(1)(2)(1) Possible Outcomes! Now we can go save the city

Permutation vs. Combination Order matters  Order matters  Order doesn’t matter  Order doesn’t matter  Permutation Combination