Compare the values of the first and last column of the table you completed in Part A.
C
c
The yellow part of the board is a circle with a hole. To find the area of a circle with a hole, first try to find the area of the same circle but without a hole.
D
d
Generate random numbers that will represent a situation of a dart landing on the green, yellow, and blue area of the dartboard.
E
e
Make a bar graph using the table you made in Part D. Each bar's height should be proportional to the corresponding relative frequency.
A
a
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
orange
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
yellow
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
green
0.25
1.5625Ï€ in^2
6.25Ï€ in^2
0.25
blue
0.25
1.5625Ï€ in^2
6.25Ï€ in^2
0.25
B
b
Example Conjecture: The probability that the spinner will stop at the given color is the same as the ratio of the area of the sector with the given color to the total area of the spinner.
C
c
Area
Probability
Green
1/9
Yellow
3/9
Blue
5/9
D
d
Example Answer:
Color
Frequency
Relative Frequency
Green
13
0.13
Yellow
29
0.29
Blue
58
0.58
E
e
Example Graph:
Practice makes perfect
a
We will starting completing the table by calculating the probabilities of all the outcomes of a spin. Let's look at the spinner.
We can see that the colors, red, orange, and yellow cover 16 of the wheel each. Color blue covers 14 of the wheel, and so does green. Let's use these values to fill in the probability column.
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
1/6
orange
1/6
yellow
1/6
green
1/4
blue
1/4
Now we can look for the sector areas. We can start by determining the angles of the sectors. The angles of the red, orange, and yellow sectors cover 180^(∘) evenly. This mean that each sector has an angle of 180^(∘)3=60^(∘). Sectors blue and green also cover 180^(∘) evenly. Therefore, their angles measures must be 180^(∘)2=90^(∘) each.
Also, we know that the radius of the circle is equal to 2.5 inches. With this information, we can calculate the areas using the formula for the area of a sector. Let's start with the red, orange, and yellow sectors. We will substitute r=2.5 inches and θ =60^(∘) into the formula.
We found that the areas of the blue and green sectors are equal to 1.5625Ï€ inches squared. Also, each area of the red, orange, and yellow sectors is equal to 1.0417Ï€ inches squared. We can note that in the given table.
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
1/6
1.0417Ï€ in^2
orange
1/6
1.0417Ï€ in^2
yellow
1/6
1.0417Ï€ in^2
green
1/4
1.5625Ï€ in^2
blue
1/4
1.5625Ï€ in^2
To find the total area, we can use the formula for the area of a circle. The given circle has a radius r=2.5 inches.
Let's write down this information in the column titled Total Area.
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
1/6
1.0417Ï€ in^2
6.25Ï€ in^2
orange
1/6
1.0417Ï€ in^2
6.25Ï€ in^2
yellow
1/6
1.0417Ï€ in^2
6.25Ï€ in^2
green
1/4
1.5625Ï€ in^2
6.25Ï€ in^2
blue
1/4
1.5625Ï€ in^2
6.25Ï€ in^2
Finally, we can calculate the values of the last column.
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
1/6≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
1.0417π/6.25π≈ 0.17
orange
1/6≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
1.0417π/6.25π≈ 0.17
yellow
1/6≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
1.0417π/6.25π≈ 0.17
green
1/4=0.25
1.5625Ï€ in^2
6.25Ï€ in^2
1.5625Ï€/6.25Ï€=0.25
blue
1/4=0.25
1.5625Ï€ in^2
6.25Ï€ in^2
1.5625Ï€/6.25Ï€=0.25
b
Before we start making conjectures, let's look at the table we completed in Part A.
Color
Probability
Sector Area
Total Area
Sector Area/Total Area
red
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
orange
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
yellow
≈ 0.17
1.0417Ï€ in^2
6.25Ï€ in^2
≈ 0.17
green
0.25
1.5625Ï€ in^2
6.25Ï€ in^2
0.25
blue
0.25
1.5625Ï€ in^2
6.25Ï€ in^2
0.25
As you can see, the values in the Probability and in the last column with the ratios of sector area to the total area are the same. Let's make a conjecture based on that fact.
The probability that the spinner will stop on the given color is the same as the ratio of the area of the sector with the given color to the total area of the spinner.
c
We are given a dartboard and we are asked to find the probability of the dart hitting each area of the board.
In Part C we made a conjecture that the area and the probability have a strong connection with each other. Therefore, to calculate the probability, we will first calculate the area of each part of the board, green, yellow, and blue.
Green Area
The green part of the dartboard is a circle with a diameter of 6 centimeters.
Therefore, the radius of the green circle is r= 62=3 centimeters and we can use the formula for the area of a circle to calculate the green area.
We found that the green area is equal to 9 π centimeters squared.
Yellow Area
The yellow area of the board is a circle with a hole inside, a doughnut shape.
To find its area, we will first find the area of the same circle but without the hole.
To find the area of the yellow circle, we need to know its radius.
Looking at the picture, we can see that the yellow circle without the hole has a radius 3 centimeters longer than the green circle. Earlier we found that the radius of the green circle is 3 centimeters. Therefore, the radius of the yellow circle is 3+3=6 centimeters.
Let's use the formula for the area of the circle with radius r=6 centimeters.
Now, we can find the area of the yellow doughnut by subtracting the area of the green circle from the area of the yellow circle.
The area of the yellow cricle is 36Ï€ and the area of the green circle was 9Ï€ . Let's subtract them!
36Ï€ - 9Ï€ =27Ï€
We found that the yellow part of the dartboard has an area of 27Ï€ centimeters squared.
Blue Area
To find the blue area we can use a similar process.
We will first find the area of the blue circle without the hole, which is also the total area of the dartboard.
To do that, we need to find the radius of the blue circle.
The radius of the blue circle without the hole is 3 centimeters longer than the yellow circle's radius. Earlier we found that the radius of the yellow circle was 6 centimeters. Therefore, the radius of the blue circle is 6+3=9 centimeters.
Let's use the formula for the area of the circle with radius r=9 centimeters.
The blue circle's area or also the total area of the dartboard is 81 π centimeters squared. Now, we can find the area of the blue part of the dartboard by subtracting the area of the yellow circle from the total area of the blue circle.
The area of the blue circle is 81Ï€ centimeters squared and the area of the yellow circle is 36Ï€ centimeters squared. Let's subtract them!
81Ï€ - 36Ï€= 45Ï€
The blue part of the dartboard has the area of 45Ï€ centimeters squared.
Predicted Probability
We found all the areas of the colored parts of the dartboard, let's summarize the results in a table.
Color
Area
Total Area
Green
9Ï€
81Ï€
Yellow
27Ï€
81Ï€
Blue
45Ï€
81Ï€
Finally, we can predict the probability of the dart landing in each area of the board. We will do that by calculating the ratio of the area of the colored part to the total area of the dartboard.
Color
Area
Total Area
Area/Total Area
Probability
Green
9Ï€
81Ï€
9Ï€/81Ï€
1/9
Yellow
27Ï€
81Ï€
27Ï€/81Ï€
3/9
Blue
45Ï€
81Ï€
45Ï€/81Ï€
5/9
d
To construct the relative-frequency table, we will need to simulate the process of throwing darts at the board. We can make a simulation using a random number generator. To determine how many numbers we need generate, let's look at the theoretical probabilities of the colors on which the dart can land that we found in Part C.
Color
Probability
Green
1/9
Yellow
3/9
Blue
5/9
The common denominator of the probabilities is 9, so we can simulate the process generating numbers 1-9. Each number will represent a dart hitting the green, yellow, or blue part of the dartboard. The numbers should be assigned proportionally to the probabilities.
Color
Probability
Represented by
Green
1/9
1
Yellow
3/9
2-4
Blue
5/9
5-9
We need to simulate one hundred dart throws, so let's use a graphing calculator to generate integers from 1 to 9 one hundred times. Start by pushing the MATH button. Then scroll right to the PRB menu and choose the randInt( option.
Since we want to generate integers from 1 to 9 one hundred times, we type 1, 9, and 100 on a graphing calculator, respectively. Then press ENTER to get one hundred outcomes.
Let's say we generated number 3. This number is in the interval 2-4, which represents the dart hitting the yellow area. So we can up the frequency of darts hitting the yellow area by one and repeat for all the other randomly generated numbers. Let's take a look at example results of the simulation.
Color
Frequency
Green
13
Yellow
29
Blue
58
Note that since this is a random simulation your results might be different. Now, to finish making the relative-frequency table we need to calculate the relative frequencies. To do that, we will divide the frequencies by the the total number of trials, 100.
look at the example results of this simulation.
Color
Frequency
Relative Frequency
Green
13
13/100=0.13
Yellow
29
29/100=0.29
Blue
58
58/100=0.58
e
We want to graph the experimental probability distribution. First, let's look at the relative frequencies we found in the previous part.
Color
Relative Frequency
Green
0.13
Yellow
0.29
Blue
0.58
We will make a bar graph using this information. Bars will represent the colors of the areas of the dart board. Each bar's height should be proportional to the corresponding relative frequency.
Since the simulation is a process affected by randomness, your graph might be a bit different.
