We throw a dart at the given board. Hitting any point inside the board is equally likely.
We will order the likelihoods of the dart landing in the given regions from least likely to most likely. To do so we will compare the of the options. We will first calculate the of the regions in each option. Then we will find the of the dart landing on these regions. Let's look at the steps that we will go through.
- We will find the probability of the green region.
- We will find the probability of the not-blue region.
- We will find the probability of the red region.
- We will find the probability of the not-yellow region.
- We will order the options.
We will start by calculating the probability of the green region.
Probability of Green Region
The area of the green region is the of two green . Since the side length of the smaller is 6 inches, the length of the base of the triangles is also 6 inches. Now we will find the height of the triangles.
By using the we will find the height of one of the green triangles.
18 in. = h + 6 in. + hFrom here we will calculate h.
Using the length of the base (6 inches) and the height (6 inches), we can find the area of one of the green triangles. Therefore, we will multiply the expression by 2.
Area of a Green Region: 2* 12(6)(6)=36in.^2
We found the area of the green region. To find the probability of the dart landing in this region, we need to evaluate the total area of the figure. The board is a square that has 18-inch side lengths, so let's find the of the board.
Area of the Figure
18^2=324in.^2
With this information we will find the probability.
P(Green)=36/324
P(Green)=1/9
P(Green)=0.11111...
P(Green)≈ 0.11
We will continue with the probability of the dart landing on a not-blue region.
Probability of Not Blue Region
Notice that the area of the not-blue region is the area of the yellow . Because one of the side lengths of the bigger square is 18 inches, the of the big yellow circle is also 18 inches. Therefore, its is 182= 9 inches.
We will substitute this value of the radius into the formula for the .
Area of yellow circle=Ï€ r^2
Area of yellow circle=Ï€ ( 9^2)
Area of yellow circle=Ï€ (81)
Area of yellow circle=81Ï€
By using the area of the yellow circle and the area of the board we will find the probability.
P(not Blue)=81Ï€/324
P(not Blue)=Ï€/4
P(not Blue)=0.78539...
P(not Blue)≈ 0.79
It is time to find the probability of the dart landing on a red region.
Probability of Red Region
We will find the area of the red region, which is the smaller circle. We are given the side length of the smaller square as 6 inches. The red circle fits the smaller square, so the diameter of the red circle is 6 inches. With this information the radius is 62= 3 inches. We are ready to find the area of the circle.
Area of inner circle=Ï€ r^2
Area of inner circle=Ï€ ( 3^2)
Area of inner circle=Ï€ (9)
Area of inner circle=9Ï€
With the information of the area of the red region and the area of the board, we will evaluate the probability.
P(Red)=9Ï€/324
P(Red)=Ï€/36
P(Red)=0.08727...
P(Red)≈ 0.09
Finally, we will find the probability for the last option.
Probability of Not Yellow Region
Note that finding the probability of landing in the yellow area is easier than finding the probability of landing in the not yellow region. Therefore we will find the of landing in the not-yellow region. We found the area of yellow region and the probability of the dart landing on the yellow region in a previous exercise.
Area of the Yellow Region = (81 π - 72) inches^2
P(yellow)≈ 0.56
We will use the complement of the event to find the probability.
P(not Yellow)=1-P(Yellow)
P(not Yellow)=1- 0.56
P(not Yellow)=0.44
We can finally order the options.
Order of Options
We know the probabilities of the options.
|
|
Events
|
Probability
|
| A.
|
Dart landing in the green region
|
0.11
|
| B.
|
Dart landing in not blue region
|
0.79
|
| C.
|
Dart landing in red region
|
0.09
|
| D.
|
Dart landing in not yellow region
|
0.44
|
From here we can order the likelihoods of the options from least likely to most likely. The probability that the dart is landing in the red region is the least likely option, whereas the probability that the dart is landing in the not-blue region is the most likely option.
C,A,D,B
