Transcript Probability & Area

Probability
& Area
1
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
=
# landing in the dartboard
area of the bull’s-eye
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.
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?
P(region B) =
area of region B
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?
P(region B) =
P(region B)
area of region B
total area of the dartboard
8
8
2
= 8 + 10 + 10 = 28 = 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 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
7
b
= 105
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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)
2
7
b
= 105
7 · 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)
2
7
b
= 105
7 · b = 2 · 105
7
7
b = 30
(Multiply to find Cross Product)
Out of 105 times, you would expect to hit Region B about 30 times.10
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
P(shaded) =
total area of the target
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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
P(shaded) =
total area of the target
12
3
P(shaded) =
=
16
4
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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
4
=
x
300
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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
4
=
x
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
4
=
x
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
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
P(region B) =
area of the target
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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
P(region B) =
area of the target
14
7
P(region B) =
=
or 0.7 or 70%
20 10
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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.
24 in
area of shaded region
P(winning) =
area of the target
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.
24 in
area of shaded region
P(winning) =
area of the target
6·6
36
1
P(winning) =
=
=
24 · 30
720
20
6 in
6 in
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
20
=
w
50
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
20
=
w
50
24 in
20 · w = 1 · 50
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
20
=
w
50
24 in
20 · w = 1 · 50
20
20
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
20
=
w
50
24 in
6 in
20 · w = 1 · 50
6 in
20
20
w = 2½
30 in
If you play 50 times you should win about 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 randomlythrown 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 randomlythrown dart landing on the shaded region. Then
predict how many of 100 darts thrown would hit
each shaded region.
(1)
½
about 50
(2)
¾
about 75
(3)
¼
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 randomlythrown 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 randomlythrown dart landing on the shaded region. Then
predict how many of 200 darts thrown would hit
each shaded region.
(1)
10/
25
2/
5
about 80
(2)
3/
4
about 150
(3)
4/
2/
6
3
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.
P(upper deck) =
Area of upper deck
Total area of 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.
22 ft
Area of upper deck
Total area of stands 43 ft
22 x 360
7920 sq ft
P(upper deck) =
=
43 x 360
23,400 sq ft
P(upper deck) =
UPPER DECK
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.
22 ft
UPPER DECK
Area of upper deck
LOWER DECK
Total area of stands 43 ft
22 x 360
7920 sq ft
360 ft
P(upper deck) =
=
43 x 360
23,400 sq ft
7920
1
P(upper deck) =
≈
or 0.33 or about 33%
23,400
3
P(upper deck) =
35
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
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
=
# landing in the dartboard
area of the bull’s-eye
total area of the dartboard
37
Probability & Area
Homework:
- Core 01 → p.___ #___, all
- Core 02 → p.___ #___, all
- Core 03 → p.___ #___, all
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