1. Created By Alan Williams
Probability & Area
Created By Alan Williams

2. 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 bull’s-eye to the total area of the dartboard

4. 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 =

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. 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) =

7. 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) = = =

8. 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 =

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

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)
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. =

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
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?

14. 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 =

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
x = 900 =

16. 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 =

17. 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.

18. 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) =

19. 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%

20. 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

21. 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

22. 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%

23. 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
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
· w = 1 · 50
24 in =
6 in
6 in
30 in

26. 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

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
· w = 1 · 50
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. (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)

29. 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

30. 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)

31. 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

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.
22 ft
UPPER DECK
43 ft
LOWER DECK
360 ft

33. 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

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

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 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%

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 bull’s-eye to the total area of the dartboard

37. 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 =

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