probability & area probability & area 1. probability & area objectives: (1) students...

ProbabilityProbability & Area & Area

1

Probability & AreaProbability & Area

ObjectivesObjectives::(1) Students will use sample space (1) Students will use sample space

to determine the probability of an to determine the probability of an event. (4.02)event. (4.02)

Essential QuestionsEssential Questions::(1) How can I use sample space to (1) How can I use sample space to

determine the probability of an determine the probability of an event?event?

(2) How can I use probability to (2) How can I use probability to make predictions?make predictions? 2


How can we use areaHow can we use areamodels to determinemodels to determinethe probability of anthe probability of anevent?event?

- Using a dartboard as an example, we can - Using a dartboard as an example, we can say the say the probability of throwing a dart probability of throwing a dart and having it hit the bull's-eyeand having it hit the bull's-eye is is equalequal to to the ratio of the area of the bull’s-eye the ratio of the area of the bull’s-eye to to the total area of the dartboardthe total area of the dartboard

3


What’s the relationshipWhat’s the relationshipbetween area andbetween area andprobability of an event?probability of an event?

Suppose you throw a large number of Suppose you throw a large number of darts at a dartboard…darts at a dartboard…

# landing in the bull’s-eye# landing in the bull’s-eye area of the bull’s-eye area of the bull’s-eye

# landing in the dartboard# landing in the dartboard total area of the total area of the dartboarddartboard

=4


Real World ExampleReal World Example:: Dartboard.Dartboard.A dartboard has three regions, A, B, and C. A dartboard has three regions, A, B, and C.

Region B has an area of 8 inRegion B has an area of 8 in22 and Regions A and Regions A and C each have an area of 10 inand C each have an area of 10 in22..

What is the probability of a randomly thrown What is the probability of a randomly thrown dart hitting Region B?dart hitting Region B?

5





area of region Barea of region B

total area of the dartboardtotal area of the dartboard

P(region B) =

6





area of region Barea of region B

total area of the dartboardtotal area of the dartboard

8 8 28 8 2

8 + 10 + 10 28 78 + 10 + 10 28 7

P(region B) =

P(region B) = = = 7




If you threw a dart 105 times, how many If you threw a dart 105 times, how many times would you expect it to hit Region B?times would you expect it to hit Region B?

(first we need to remember that from the previous question, (first we need to remember that from the previous question, there is a 2/7 chance of hitting Region B if we randomly there is a 2/7 chance of hitting Region B if we randomly throw a dart)throw a dart)

22 b b

77 105 105

=

8




If you threw a dart 105 times, how many If you threw a dart 105 times, how many times would you expect it to hit Region B?times would you expect it to hit Region B?

(first we need to remember that from the previous question, (first we need to remember that from the previous question, there is a 2/7 chance of hitting Region B if we randomly throw there is a 2/7 chance of hitting Region B if we randomly throw a dart)a dart)

22 b b 7 7 ·· b = 2b = 2 ·· 105 105 (Multiply to find Cross (Multiply to find Cross Product)Product)

77 105 105

=

9




If you threw a dart 105 times, how many times If you threw a dart 105 times, how many times would you expect it to hit Region B?would you expect it to hit Region B?

(first we need to remember that from the previous question, there (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)is a 2/7 chance of hitting Region B if we randomly throw a dart)

22 b b 7 7 ·· b = 2b = 2 ·· 105 105 (Multiply to find Cross (Multiply to find Cross Product)Product)

77 105 105 7 7 7 7 b = 30b = 30OOut of 105 times, you would expect to hit Region B about 30 ut of 105 times, you would expect to hit Region B about 30

times.times.

=

10


Example 1Example 1:: Finding probability Finding probability using area.using area.

What is the probability that a randomly thrown What is the probability that a randomly thrown dart will land in the shaded region?dart will land in the shaded region?

number of shaded regionnumber of shaded region

total area of the targettotal area of the target

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P(shaded) P(shaded) ==



What is the probability that a randomly thrown What is the probability that a randomly thrown dart will land in the shaded region?dart will land in the shaded region?

number of shaded regionnumber of shaded region

total area of the targettotal area of the target

1212 3 3

1616 4 4

12

P(shaded) P(shaded) ==

P(shaded) =P(shaded) = = =



If Mr. Williams randomly drops 300 pebbles onto If Mr. Williams randomly drops 300 pebbles onto the squares, how many should land in the the squares, how many should land in the shaded region?shaded region?

13




33 xx

44 300 300

14

=




33 xx

44 300 300 4x = 900 4x = 900

15

=




33 xx

44 300 300 4x = 900 4x = 900

4 44 4

x = 225 pebblesx = 225 pebbles

16

=


Example 2Example 2:: Carnival Games.Carnival Games.Steve and his family are at the fair. Walking Steve and his family are at the fair. Walking

around Steve’s boys Tom and Jerry ask if they around Steve’s boys Tom and Jerry ask if they can play a game where you toss a coin and try can play a game where you toss a coin and try to have it land on a certain area. If it lands in to have it land on a certain area. If it lands in that area you win a prize. Find the probability that area you win a prize. Find the probability that Tom and Jerry will win a prize.that Tom and Jerry will win a prize.

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area of shaded regionarea of shaded region

area of the targetarea of the target

P(region B) =

18






1414 7 7

2020 1010

P(region B) = P(region B) = = or 0.7 or 70%

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Example 3Example 3:: Carnival Games 2.Carnival Games 2.A carnival game involves throwing a bean bag at A carnival game involves throwing a bean bag at

a target. If the bean bag hits the shaded a target. If the bean bag hits the shaded portion of the target, the player wins. Find the portion of the target, the player wins. Find the probability that a player will win. Assume it is probability that a player will win. Assume it is equally likely to hit anywhere on the target.equally likely to hit anywhere on the target.

6 in

6 in

30 in

24 in

20






6 in

6 in

30 in

24 in

P(winning) =

21






66 ·· 6 6 36 36 11

2424 ·· 30 720 30 720 2020

6 in

6 in

30 in

24 in

P(winning) = P(winning) = = =

or 0.05 or 5% 22


Example 4Example 4:: Probability & Probability & Predictions.Predictions.

From the previous example we determined there From the previous example we determined there was a 1/20 or 5% chance of the bean bag was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. landing in the shaded portion of the target. Predict how many times you would win the Predict how many times you would win the carnival game if you played 50 times.carnival game if you played 50 times.

6 in

6 in

30 in

24 in

23




1 w w is # of wins1 w w is # of wins

20 20 50 50 number of playsnumber of plays6 in

6 in

30 in

24 in

=

24




1 w1 w

20 20 50 2050 20 ·· w = 1w = 1 ·· 5050

6 in

6 in

30 in

24 in

=

25




1 w1 w

20 20 50 2050 20 ·· w = 1w = 1 ·· 5050

20 20 2020

6 in

6 in

30 in

24 in

=

26


Example 4Example 4:: Probability & Predictions.Probability & Predictions.From the previous example we determined there From the previous example we determined there

was a 1/20 or 5% chance of the bean bag landing was a 1/20 or 5% chance of the bean bag landing in the shaded portion of the target. Predict how in the shaded portion of the target. Predict how many times you would win the carnival game if many times you would win the carnival game if you played 50 times.you played 50 times.

1 w1 w

20 20 50 2050 20 ·· w = 1w = 1 ·· 5050

20 20 2020

w = 2½ w = 2½

If you play 50 times you should win about 3.If you play 50 times you should win about 3.

6 in

6 in

30 in

24 in

=

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Guided PracticeGuided Practice:: Dartboards.Dartboards.Each figure represents a dartboard. If it is Each figure represents a dartboard. If it is

equally likely that a dart will land anywhere on equally likely that a dart will land anywhere on the dartboard, find the probability of a the dartboard, find the probability of a randomly-thrown dart landing on the shaded randomly-thrown dart landing on the shaded region. Then predict how many of 100 darts region. Then predict how many of 100 darts thrown would hit each shaded region.thrown would hit each shaded region.

(1)(1) (2)(2) (3)(3)

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Guided PracticeGuided Practice:: Dartboards.Dartboards.Each figure represents a dartboard. If it is Each figure represents a dartboard. If it is

equally likely that a dart will land anywhere on equally likely that a dart will land anywhere on the dartboard, find the probability of a the dartboard, find the probability of a randomly-thrown dart landing on the shaded randomly-thrown dart landing on the shaded region. Then predict how many of 100 darts region. Then predict how many of 100 darts thrown would hit each shaded region.thrown would hit each shaded region.

(1) (1) ½½ (2)(2) ¾ ¾ (3)(3) ¼ ¼

about 50about 50 about 75 about 75 about about 2525

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Independent PracticeIndependent Practice:: Complete Each Complete Each Example.Example.

Each figure represents a dartboard. If it is Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on equally likely that a dart will land anywhere on the dartboard, find the probability of a the dartboard, find the probability of a randomly-thrown dart landing on the shaded randomly-thrown dart landing on the shaded region. Then predict how many of 200 darts region. Then predict how many of 200 darts thrown would hit each shaded region.thrown would hit each shaded region.

(1) (1) (2)(2) (3)(3)

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Independent PracticeIndependent Practice:: Complete Each Complete Each Example.Example.

Each figure represents a dartboard. If it is equally Each figure represents a dartboard. If it is equally likely that a dart will land anywhere on the likely that a dart will land anywhere on the dartboard, find the probability of a randomly-dartboard, find the probability of a randomly-thrown dart landing on the shaded region. Then thrown dart landing on the shaded region. Then predict how many of 200 darts thrown would hit predict how many of 200 darts thrown would hit each shaded region.each shaded region.

(1) (1) 1010//2525 (2)(2) 33//44 (3)(3) 44//66

22//55

22//33

about 80about 80 about 150 about 150 about about 133133

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Real World ExampleReal World Example:: T-Shirts.T-Shirts.A cheerleading squad plans to throw t-shirts into A cheerleading squad plans to throw t-shirts into

the stands using a sling shot. Find the the stands using a sling shot. Find the probability that a t-shirt will land in the upper probability that a t-shirt will land in the upper deck of the stands. Assume it is equally likely deck of the stands. Assume it is equally likely for a shirt to land anywhere in the stands.for a shirt to land anywhere in the stands.

LOWER DECK

UPPER DECK

22 ft

43 ft

360 ft

32




Area of upper deckArea of upper deck

Total area of standsTotal area of stands

LOWER DECK

UPPER DECK

22 ft

43 ft

360 ft

P(upper deck) P(upper deck) ==

33






22 x 360 7920 sq ft22 x 360 7920 sq ft

43 x 360 43 x 360 23,400 sq ft23,400 sq ft

LOWER DECK

UPPER DECK

22 ft

43 ft

360 ft


P(upper deck) = =P(upper deck) = =

34






22 x 360 7920 sq ft22 x 360 7920 sq ft

43 x 360 43 x 360 23,400 sq ft23,400 sq ft

7920 17920 1

23,400 23,400 33

LOWER DECK

UPPER DECK

22 ft

43 ft

360 ft


P(upper deck) = =P(upper deck) = =

P(upper deck) = ≈P(upper deck) = ≈ or 0.33 or about 33%35


How can we use areaHow can we use areamodels to determinemodels to determinethe probability of anthe probability of anevent?event?

- Using a dartboard as an example, we can - Using a dartboard as an example, we can say the say the probability of throwing a dart probability of throwing a dart and having it hit the bull's-eyeand having it hit the bull's-eye is is equalequal to to the ratio of the area of the bull’s-eye the ratio of the area of the bull’s-eye to to the total area of the dartboardthe total area of the dartboard

36


What’s the relationshipWhat’s the relationshipbetween area andbetween area andprobability of an event?probability of an event?

Suppose you throw a large number of Suppose you throw a large number of darts at a dartboard…darts at a dartboard…

# landing in the bull’s-eye# landing in the bull’s-eye area of the bull’s-eye area of the bull’s-eye

# landing in the dartboard# landing in the dartboard total area of the total area of the dartboarddartboard

=37

HomeworkHomework::- Core 01 - Core 01 → p.___ #___, all→ p.___ #___, all

- Core 02 - Core 02 → p.___ #___, all→ p.___ #___, all

- Core 03 - Core 03 → p.___ #___, all→ p.___ #___, all


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probability & area probability & area 1. probability & area objectives: (1) students...

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