Unit 8. Day 6.

Today we will review “tree diagrams” You may have already seen tree diagrams (maybe not applied to probability) This easy idea will be used to remind us of the a “trick” Then we will use tree diagrams & the “trick” to help us find probabilities of compound events. CCSS.MATH.CONTENT.7.SP.C.8: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Example A: Gabi is packing for a trip Example A: Gabi is packing for a trip. She has 4 tops, 3 bottoms, and 2 pairs of shoes. How many different outfits can she make?

There must be a better way! Example A: Gabi is packing for a trip. She has 4 tops, 3 bottoms, and 2 pairs of shoes. How many different outfits can she make? 𝑆= 𝑇1 𝐵1 𝑆1 𝑇2 𝐵1 𝑆1 𝑇1 𝐵1 𝑆2 𝑇2 𝐵1 𝑆2 𝑇1 𝐵2 𝑆1 𝑇1 𝐵2 𝑆2 𝑇1 𝐵3 𝑆1 𝑇1 𝐵3 𝑆2 This is crazy! There must be a better way!

Example A: Gabi is packing for a trip Example A: Gabi is packing for a trip. She has 4 tops, 3 bottoms, and 2 pairs of shoes. How many different outfits can she make? 𝑇1 𝑇2 𝑇3 𝑇4 𝐵1 𝐵3 𝐵1 𝐵2 𝐵3 𝐵2 𝐵1 𝐵2 𝐵3 𝐵1 𝐵2 𝐵3 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2 𝑆1 𝑆2

Example A: Gabi is packing for a trip Example A: Gabi is packing for a trip. She has 4 tops, 3 bottoms, and 2 pairs of shoes. How many different outfits can she make? Multiplying “trick”: 4 ∙ 3 ∙ 2 = 24

Strawberry Strawberry Example B: An ice cream shop has 2 types of cones, 4 flavors, & 2 toppings. How many different ice cream cones can be made ? Waffle Mango Chocolate Waffle Mango Strawberry Waffle Vanilla Chocolate Waffle Vanilla Strawberry Waffle Mint CC Chocolate Waffle Mint CC Strawberry Waffle Chocolate Chocolate Waffle Chocolate Strawberry Sugar Mango Chocolate Sugar Mango Strawberry Sugar Vanilla Chocolate Sugar Vanilla Strawberry Sugar Mint CC Chocolate Sugar Mint CC Strawberry Sugar Chocolate Chocolate Sugar Chocolate Strawberry

Multiplication: 2 ∙ 4 ∙ 2 = 16 Waffle Sugar Mango Vanilla Mint CC Chocolate Mango Strawberry Chocolate Vanilla Strawberry Waffle Chocolate Mint CC Strawberry Multiplication: Chocolate Chocolate Strawberry 2 ∙ 4 ∙ 2 = 16 Chocolate Mango Strawberry Chocolate Vanilla Sugar Strawberry Chocolate Mint CC Strawberry Chocolate Chocolate Strawberry

Example C: A car comes with the choice of gas or diesel engines, automatic or manual transmissions, and cloth or leather seats. The car has five different colors: white, green, red, silver, black. How many different cars are there? w g r s b w g r s w b g r w s b g r s w b g r s w b g r s w b Multiplication: g r s w b g 2 ∙ 2 ∙ 2 = 40 r ∙ 5 s b

Examples A, B, and C were not random/chance events. In 7th grade we use tree diagrams to find sample spaces & probabilities of chance events

Example D: Carnival Game. 𝑆= 𝑅 𝐵 𝐵 𝑅 𝐵 𝑅 𝐵 𝑅 𝑅 𝐵 𝑌 𝐺 𝐵 𝑅 𝑌 𝐺

Example D: Carnival Game 𝑅𝑒𝑑 𝐵𝑙𝑢𝑒 𝑅𝑒𝑑 𝐵𝑙𝑢𝑒 𝑌𝑒𝑙𝑙𝑜𝑤 𝐺𝑟𝑒𝑒𝑛 𝑅𝑒𝑑 𝐵𝑙𝑢𝑒 𝑌𝑒𝑙𝑙𝑜𝑤 𝐺𝑟𝑒𝑒𝑛

Example E: A dice is rolled twice? 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 9 1 4 𝑃 𝑏𝑜𝑡ℎ 𝑟𝑜𝑙𝑙𝑠 𝑎𝑟𝑒 𝑜𝑑𝑑 = = 36

Example F: A coin is flipped 4 times Example F: A coin is flipped 4 times. What is the probability of getting at least three heads? 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 5 𝑃 𝐻≥3 = 16