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4.3 Counting Techniques Prob & Stats Tree Diagrams When calculating probabilities, you need to know the total outcomes number of _____________ in the ______________. sample space Tree Diagrams Example Use a TREE DIAGRAM to list the sample space of 2 coin flips. Sample Space H H Now you could get… If you got H T On the first flip you could get….. YOU H you could NowIf you got Tget… T T Tree Diagram Example Mr. Arnold’s Closet 3 Shirts 2 Pants 2 Pairs of Shoes Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 3 4 Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 3 4 5 6 Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 3 4 5 6 7 8 Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 3 4 5 6 7 8 9 10 Dress Mr. Arnold List all of Mr. Arnold’s outfits 1 2 3 4 5 6 7 8 9 10 11 12 Dress Mr. Arnold If Mr. Arnold picks an outfit with his eyes List all of Mr. Arnold’s outfits 1 closed……. 2 P(brown shoe) = 3 6/12 1/2 4 P(polo) = 5 6 1/3 4/12 7 P(lookin’ cool) = 8 9 10 11 1 12 Multiplication Rule of Counting The size of the sample space is denominator the ___________ of our probability So we don’t always need to know what each outcome is, just the number of outcomes. 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. Arnold Mr. Reed had 3 EVENTS 2 shoes 2 pants 3 shirts How many outcomes are there for = 12 OUTFITS 2(2)(3)EACH EVENT? Permutations Sometimes we are concerned with how many ways a group of arranged objects can be __________. •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 Wonder Woman’s invisible plane has 3 Example: chairs. There are 3 people who need a lift. 3 People, 3 Chairs How many seating options are there? Think of each chair as Wonder Woman driving 6 Seating Options! an EVENT Batman driving 3 2 1 Now that ways is filled? How manythe 1st could the Now the first 2 are filled. How = options for many 6 OPTIONS 3(2)(1)many ways to fill ndrd? st How 1 chair be filled? 2 3 ? Superman driving Example: The batmobile has 5 chairs. There are 5 people who need a lift. 5 People, 5 Chairs How many seating options are there? 5 4 3 2 1 =120 Seating Options Multiply!! This is a PERMUTATION of 5 objects Commercial Break: FACTORIAL denoted with ! 5! Multiply all integers ≤ the number 5! = 5(4)(3)(2)(1) = 120 0! = 1 1! = 1 Calculate 6! 6! = 6(5)(4)(3)(2)(1) = 720 What is 6! / 5!? Commercial Break: FACTORIAL denoted with ! 5! Multiply all integers ≤ the number 0! = 1 1! = 1 Calculate 6! What is 6! / 5!? 6(5)(4)(3)(2)(1) 5(4)(3)(2)(1) =6 Example: The batmobile has 5 chairs. There are 5 people who need a lift. 5 People, 5 Chairs How many seating options are there? 5! 5 4 3 2 1 =120 Seating Options 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 Think of the possibilities! arrange them 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 You have 3 EVENTS? How many outcomes for this event! 4 Now someone is on FIRST 5 You have to choose 3 AND arrange them 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 n objects You select r objects This is the number of ways you could select and arrange in order: Another common notation for a permutation is nPr Softball Permutation Revisited 5! 5(4)(3)(2)(1) 5 people to choose from n= r = 3 spots to fill (5 – 2! 3)! 2(1) 5(4)(3) = 120 POSSIBLITIES You have to choose 3 AND arrange them Combinations Sometimes, we are only concerned with selecting a group and not the order in which they are selected. A combination gives the number of ways to select a sample of r objects from a group of size n. 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? NOPE Does it matter who is selected last? NOPE Combination: Duty Calls Let’s look at the drawing possibilities STOP! This is a waste of time These are all them We’ll countthewho gotas The monster doesn’t care SAME: drawn ONE OUTCOMEfirst. All these outcomes = same people We’ll count them as These are all the SAME: pounding his face The monster doesn’t care who got drawn ONE OUTCOME first. All these outcomes = same people pounding his face Combination: Duty Calls Okay, let’s consider other outcomes 10 Possible Outcomes! Combination Formula You have n objects You want a group of r objects You DON’T CARE what order they are selected in Combinations are also denoted nCr Read “n choose r” Duty Calls: Revisited n = 5 people toDOESN’T MATTER ORDER choose from r = 3 spots to fill 5(4)(3)(2)(1) 5! 20 2 3!(2)! 3!(5 - 3)! 3(2)(1)(2)(1) 10 Possible Outcomes! Now we can go save the city Permutation vs. Combination Order matters Permutation Order doesn’t matter Combination

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Problem Solving, Addition and Subtraction, how to, Lesson 1, Lesson 2, factor 2, power comparisons, Long Division, Counting Money, shopping cart

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posted: | 5/18/2011 |

language: | English |

pages: | 40 |

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