Probability

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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?

3
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 bulls-eye to the total area of the
  dartboard

4
Probability Area

• Whats the relationship
• between area and
• probability of an event?
• Suppose you throw a large number of darts at a
  dartboard
• landing in the bulls-eye area of
  the bulls-eye
• landing in the dartboard total area of
  the dartboard

5
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?

6
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?
• area of region B
• total area of the dartboard

P(region B)
7
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?
• area of region B
• total area of the dartboard
• 8 8 2
• 8 10 10 28 7

P(region B)
P(region B)
8
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)
• 2 b
• 7 105

9
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)
• 2 b 7 b 2 105 (Multiply to
  find Cross Product)
• 7 105

10
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)
• 2 b 7 b 2 105 (Multiply to
  find Cross Product)
• 7 105 7 7
• b 30
• Out of 105 times, you would expect to hit Region
  B about 30 times.

11
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)
12
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
• 12 3
• 16 4

P(shaded)
P(shaded)
13
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
• 4 300

15
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

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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
• x 225 pebbles

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Probability Area

• Example 2 Carnival Games.
• Steve and his family are at the fair. Walking
  around Steves 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.

18
Probability Area

• Example 2 Carnival Games.
• Steve and his family are at the fair. Walking
  around Steves 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)
19
Probability Area

• Example 2 Carnival Games.
• Steve and his family are at the fair. Walking
  around Steves 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 6 36 1
• 24 30 720 20

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
24
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
• 20 50 number of plays

24 in

6 in
6 in
30 in
25
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
• 20 50 20 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
• 20 50 20 w 1 50
• 20 20

24 in

6 in
6 in
30 in
27
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
• 20 50 20 w 1 50
• 20 20
• w 2½
• If you play 50 times you should win about 3.

24 in

6 in
6 in
30 in
28
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)

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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

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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)

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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
• about 80 about 150 about 133

32
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.

UPPER DECK
22 ft
LOWER DECK
43 ft
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

UPPER DECK
22 ft
P(upper deck)
LOWER DECK
43 ft
360 ft
34
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

UPPER DECK
22 ft
P(upper deck)
LOWER DECK
43 ft
360 ft
P(upper deck)
35
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
• 23,400 3

UPPER DECK
22 ft
P(upper deck)
LOWER DECK
43 ft
360 ft
P(upper deck)
P(upper deck) or 0.33
or about 33
36
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 bulls-eye to the total area of the
  dartboard

37
Probability Area

• Whats the relationship
• between area and
• probability of an event?
• Suppose you throw a large number of darts at a
  dartboard
• landing in the bulls-eye area of
  the bulls-eye
• landing in the dartboard total area of
  the dartboard

38
Probability Area

• Homework
• - Core 01 ? p.___ ___, all
• - Core 02 ? p.___ ___, all
• - Core 03 ? p.___ ___, all