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Probability & Area Probability & Area 1

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Probability & Area Objectives: (1) Students will use sample space to determine the probability of an event. (4.02) Essential Questions: (1) How can I use sample space to determine the probability of an event? (2) How can I use probability to make predictions? 2

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Probability & Area How can we use area models to determine the probability of an event? - Using a dartboard as an example, we can say the probability of throwing a dart and having it hit the bull's-eye is equal to the ratio of the area of the bull’s-eye to the total area of the dartboard 3

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Probability & Area What’s the relationship between area and probability of an event? Suppose you throw a large number of darts at a dartboard… # landing in the bull’s-eye area of the bull’s-eye # landing in the dartboard total area of the dartboard = 4

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. What is the probability of a randomly thrown dart hitting Region B? 5

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. What is the probability of a randomly thrown dart hitting Region B? area of region B area of region B total area of the dartboard total area of the dartboard P(region B) = 6

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. What is the probability of a randomly thrown dart hitting Region B? area of region B area of region B total area of the dartboard total area of the dartboard 8 8 2 8 8 2 8 + 10 + 10 28 7 P(region B) = P(region B) = = = 7

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. If you threw a dart 105 times, how many times would you expect it to hit Region B? (first we need to remember that from the previous question, there is a 2/7 chance of hitting Region B if we randomly throw a dart) 2 b 7 105 = 8

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. If you threw a dart 105 times, how many times would you expect it to hit Region B? (first we need to remember that from the previous question, there is a 2/7 chance of hitting Region B if we randomly throw a dart) 2 b 7 · b = 2 · 105 (Multiply to find Cross Product) 7 105 = 9

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Probability & Area Real World Example: Dartboard. A dartboard has three regions, A, B, and C. Region B has an area of 8 in 2 and Regions A and C each have an area of 10 in 2. If you threw a dart 105 times, how many times would you expect it to hit Region B? (first we need to remember that from the previous question, there is a 2/7 chance of hitting Region B if we randomly throw a dart) 2 b 7 · b = 2 · 105 (Multiply to find Cross Product) 7 105 7 7 b = 30 b = 30 O ut of 105 times, you would expect to hit Region B about 30 times. = 10

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Probability & Area Example 1: Finding probability using area. What is the probability that a randomly thrown dart will land in the shaded region? number of shaded region number of shaded region total area of the target total area of the target 11 P(shaded) =

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Probability & Area Example 1: Finding probability using area. What is the probability that a randomly thrown dart will land in the shaded region? number of shaded region number of shaded region total area of the target total area of the target 12 3 12 3 16 4 12 P(shaded) = P(shaded) = =

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Probability & Area Example 1: Finding probability using area. If Mr. Williams randomly drops 300 pebbles onto the squares, how many should land in the shaded region? 13

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Probability & Area Example 1: Finding probability using area. If Mr. Williams randomly drops 300 pebbles onto the squares, how many should land in the shaded region? 3 x 4 300 14 =

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Probability & Area Example 1: Finding probability using area. If Mr. Williams randomly drops 300 pebbles onto the squares, how many should land in the shaded region? 3 x 4 300 4x = 900 15 =

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Probability & Area Example 1: Finding probability using area. If Mr. Williams randomly drops 300 pebbles onto the squares, how many should land in the shaded region? 3 x 4 300 4x = 900 4 4 4 4 x = 225 pebbles 16 =

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Probability & Area Example 2: Carnival Games. Steve and his family are at the fair. Walking around Steve’s boys Tom and Jerry ask if they can play a game where you toss a coin and try to have it land on a certain area. If it lands in that area you win a prize. Find the probability that Tom and Jerry will win a prize. 17

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Probability & Area Example 2: Carnival Games. Steve and his family are at the fair. Walking around Steve’s boys Tom and Jerry ask if they can play a game where you toss a coin and try to have it land on a certain area. If it lands in that area you win a prize. Find the probability that Tom and Jerry will win a prize. area of shaded region area of shaded region area of the target area of the target P(region B) = 18

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Probability & Area Example 2: Carnival Games. Steve and his family are at the fair. Walking around Steve’s boys Tom and Jerry ask if they can play a game where you toss a coin and try to have it land on a certain area. If it lands in that area you win a prize. Find the probability that Tom and Jerry will win a prize. area of shaded region area of shaded region area of the target area of the target 14 7 14 7 20 10 20 10 P(region B) = P(region B) = = or 0.7 or 70% 19

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Probability & Area Example 3: Carnival Games 2. A carnival game involves throwing a bean bag at a target. If the bean bag hits the shaded portion of the target, the player wins. Find the probability that a player will win. Assume it is equally likely to hit anywhere on the target. 6 in 30 in 24 in 20

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Probability & Area Example 3: Carnival Games 2. A carnival game involves throwing a bean bag at a target. If the bean bag hits the shaded portion of the target, the player wins. Find the probability that a player will win. Assume it is equally likely to hit anywhere on the target. area of shaded region area of the target area of the target 6 in 30 in 24 in P(winning) = 21

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Probability & Area Example 3: Carnival Games 2. A carnival game involves throwing a bean bag at a target. If the bean bag hits the shaded portion of the target, the player wins. Find the probability that a player will win. Assume it is equally likely to hit anywhere on the target. area of shaded region area of the target area of the target 6 · 6 36 1 6 · 6 36 1 24 · 30 720 20 6 in 30 in 24 in P(winning) = P(winning) = = = or 0.05 or 5% 22

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Probability & Area Example 4: Probability & Predictions. From the previous example we determined there was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how many times you would win the carnival game if you played 50 times. 6 in 30 in 24 in 23

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Probability & Area Example 4: Probability & Predictions. From the previous example we determined there was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how many times you would win the carnival game if you played 50 times. 1 w w is # of wins 1 w w is # of wins 20 50 number of plays 6 in 30 in 24 in = 24

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Probability & Area Example 4: Probability & Predictions. From the previous example we determined there was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how many times you would win the carnival game if you played 50 times. 1 w 1 w 20 50 20 · w = 1 · 50 6 in 30 in 24 in = 25

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Probability & Area Example 4: Probability & Predictions. From the previous example we determined there was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how many times you would win the carnival game if you played 50 times. 1 w 1 w 20 50 20 · w = 1 · 50 20 20 20 20 6 in 30 in 24 in = 26

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Probability & Area Example 4: Probability & Predictions. From the previous example we determined there was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how many times you would win the carnival game if you played 50 times. 1 w 1 w 20 50 20 · w = 1 · 50 20 20 20 20 w = 2½ w = 2½ If you play 50 times you should win about 3. 6 in 30 in 24 in = 27

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Probability & Area Guided Practice: Dartboards. Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on the dartboard, find the probability of a randomly- thrown dart landing on the shaded region. Then predict how many of 100 darts thrown would hit each shaded region. (1)(2)(3) 28

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Probability & Area Guided Practice: Dartboards. Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on the dartboard, find the probability of a randomly- thrown dart landing on the shaded region. Then predict how many of 100 darts thrown would hit each shaded region. (1) ½ (2) ¾ (3) ¼ about 50 about 75 about 25 about 50 about 75 about 25 29

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Probability & Area Independent Practice: Complete Each Example. Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on the dartboard, find the probability of a randomly- thrown dart landing on the shaded region. Then predict how many of 200 darts thrown would hit each shaded region. (1) (2) (3) 30

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Probability & Area Independent Practice: Complete Each Example. Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on the dartboard, find the probability of a randomly- thrown dart landing on the shaded region. Then predict how many of 200 darts thrown would hit each shaded region. (1) 10 / 25 (2) 3 / 4 (3) 4 / 6 2 / 5 2 / 3 2 / 5 2 / 3 about 80 about 150 about 133 about 80 about 150 about 133 31

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Probability & Area Real World Example: T-Shirts. A cheerleading squad plans to throw t-shirts into the stands using a sling shot. Find the probability that a t-shirt will land in the upper deck of the stands. Assume it is equally likely for a shirt to land anywhere in the stands. LOWER DECK UPPER DECK 22 ft 43 ft 360 ft 32

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Probability & Area Real World Example: T-Shirts. A cheerleading squad plans to throw t-shirts into the stands using a sling shot. Find the probability that a t-shirt will land in the upper deck of the stands. Assume it is equally likely for a shirt to land anywhere in the stands. Area of upper deck Total area of stands LOWER DECK UPPER DECK 22 ft 43 ft 360 ft P(upper deck) = P(upper deck) = 33

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Probability & Area Real World Example: T-Shirts. A cheerleading squad plans to throw t-shirts into the stands using a sling shot. Find the probability that a t-shirt will land in the upper deck of the stands. Assume it is equally likely for a shirt to land anywhere in the stands. Area of upper deck Total area of stands 22 x 360 7920 sq ft 43 x 360 23,400 sq ft LOWER DECK UPPER DECK 22 ft 43 ft 360 ft P(upper deck) = P(upper deck) = P(upper deck) = = P(upper deck) = = 34

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Probability & Area Real World Example: T-Shirts. A cheerleading squad plans to throw t-shirts into the stands using a sling shot. Find the probability that a t-shirt will land in the upper deck of the stands. Assume it is equally likely for a shirt to land anywhere in the stands. Area of upper deck Total area of stands 22 x 360 7920 sq ft 43 x 360 23,400 sq ft 7920 1 7920 1 23,400 3 23,400 3 LOWER DECK UPPER DECK 22 ft 43 ft 360 ft P(upper deck) = P(upper deck) = P(upper deck) = = P(upper deck) = = P(upper deck) = ≈ P(upper deck) = ≈ or 0.33 or about 33% 35

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Probability & Area How can we use area models to determine the probability of an event? - Using a dartboard as an example, we can say the probability of throwing a dart and having it hit the bull's-eye is equal to the ratio of the area of the bull’s-eye to the total area of the dartboard 36

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Probability & Area What’s the relationship between area and probability of an event? Suppose you throw a large number of darts at a dartboard… # landing in the bull’s-eye area of the bull’s-eye # landing in the dartboard total area of the dartboard = 37

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Homework: - Core 01 → p.___ #___, all - Core 02 → p.___ #___, all - Core 03 → p.___ #___, all Probability & Area 38

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