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**Created By Alan Williams**

Probability & Area Created By Alan Williams

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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?

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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

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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 =

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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 in2 and Regions A and C each have an area of 10 in2. What is the probability of a randomly thrown dart hitting Region B?

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**Probability & Area Real World Example: Dartboard. P(region B) =**

A dartboard has three regions, A, B, and C. Region B has an area of 8 in2 and Regions A and C each have an area of 10 in2. What is the probability of a randomly thrown dart hitting Region B? area of region B total area of the dartboard P(region B) =

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**Probability & Area Real World Example: Dartboard. P(region B) =**

A dartboard has three regions, A, B, and C. Region B has an area of 8 in2 and Regions A and C each have an area of 10 in2. What is the probability of a randomly thrown dart hitting Region B? area of region B total area of the dartboard P(region B) = P(region B) = = =

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**Probability & Area = Real World Example: Dartboard. 2 b 7 105**

A dartboard has three regions, A, B, and C. Region B has an area of 8 in2 and Regions A and C each have an area of 10 in2. 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) b =

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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 in2 and Regions A and C each have an area of 10 in2. 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) b · b = 2 · 105 (Multiply to find Cross Product) =

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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 in2 and Regions A and C each have an area of 10 in2. 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) b · b = 2 · 105 (Multiply to find Cross Product) b = 30 Out of 105 times, you would expect to hit Region B about 30 times. =

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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 total area of the target 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 total area of the target 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?

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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 =

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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 x = 900 =

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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 x = 900 x = 225 pebbles =

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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.

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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 the target P(region B) =

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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 the target 14 7 20 10 P(region B) = P(region B) = = or 0.7 or 70%

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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. 24 in 6 in 6 in 30 in

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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 24 in 6 in P(winning) = 6 in 30 in

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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 6 · 24 · 24 in 6 in P(winning) = 6 in P(winning) = = = 30 in or 0.05 or 5%

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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. 24 in 6 in 6 in 30 in

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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 number of plays 24 in = 6 in 6 in 30 in

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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 = 1 · 50 24 in = 6 in 6 in 30 in

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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 = 1 · 50 24 in = 6 in 6 in 30 in

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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 = 1 · 50 w = 2½ If you play 50 times you should win about 3. 24 in = 6 in 6 in 30 in

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**Probability & Area Guided Practice: Dartboards. (1) (2) (3)**

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)

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**Probability & Area Guided Practice: Dartboards. (1) ½ (2) ¾ (3) ¼**

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 about about 25

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**Probability & Area (1) (2) (3)**

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)

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**Probability & Area (1) 10/25 (2) 3/4 (3) 4/6 2/5 2/3**

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) /25 (2) 3/4 (3) 4/6 2/ /3 about about about 133

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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. 22 ft UPPER DECK 43 ft LOWER DECK 360 ft

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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 ft UPPER DECK P(upper deck) = 43 ft LOWER DECK 360 ft

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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 sq ft 43 x ,400 sq ft 22 ft UPPER DECK P(upper deck) = 43 ft LOWER DECK P(upper deck) = = 360 ft

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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 sq ft 43 x ,400 sq ft 23, 22 ft UPPER DECK P(upper deck) = 43 ft LOWER DECK P(upper deck) = = 360 ft P(upper deck) = ≈ or 0.33 or about 33%

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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

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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 =

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**Probability & Area Homework: - Core 01 → p.___ #___, all**

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