Counting Techniques Prob & Stats

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

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

Tree Diagram Example Mr. Milano’s Closet 3 Shirts 2 Pants 2 Pairs of Shoes

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

Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4

Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4 5 6

Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8

Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8 9 10

Dress Mr. Milano List all of Mr. Milano’s outfits 1 2 3 4 5 6 7 8 9 10 11 12

1 6/12 1/2 1/3 4/12 Dress Mr. Milano List all of Mr. Milano’s outfits If Mr. Milano picks an outfit with his eyes closed……. List all of Mr. Milano’s outfits 1 2 P(brown shoe) = 6/12 3 1/2 4 P(polo) = 5 1/3 4/12 6 7 P(lookin’ cool) = 1 8 9 10 11 12

Multiplication Rule of Counting 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. denominator number

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

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

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

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

Example: 5 People, 5 Chairs The batmobile has 5 chairs. There are 5 people who need a lift. How many seating options are there? =120 Seating Options 5 4 3 2 1 Multiply!! This is a PERMUTATION of 5 objects

Commercial Break: FACTORIAL shown with !  Multiply all integers ≤ the number  0! = 1! = Calculate 6! What is 6! / 5!? 5! 5! = 5(4)(3)(2)(1) = 120 1 1 6! = 6(5)(4)(3)(2)(1) = 720

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

Example: 5 People, 5 Chairs The batmobile has 5 chairs. There are 5 people who need a lift. How many seating options are there? 5! =120 Seating Options 5 4 3 2 1 Multiply!! This is a PERMUTATION of 5 objects

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

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

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

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

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

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

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

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

4 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 to choose 3 AND arrange them

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

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

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

Combinations 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. selecting not the order combination select a sample

Combination: Duty Calls There is an evil monster threatening the city. The mayor calls the Justice League. He requests that 3 members be sent to combat the menace. 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 last? NOPE NOPE

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

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

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

Duty Calls: Revisited 5! 3!(5 - 3)! 3!(2)! 5(4)(3)(2)(1) 3(2)(1)(2)(1) n = people to choose from ORDER DOESN’T MATTER r = 3 spots to fill 5(4)(3)(2)(1) 5! 20 2 3!(5 - 3)! 3!(2)! 3(2)(1)(2)(1) 10 Possible Outcomes! Now we can go save the city

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