A square dart board has four dark circular regions of radius 3 inches as shown in the design above. Each point on the dart board is equally likely to be hit by a dart that hits the board. What is the probability that a dart that hits the board will hit one of the circular regions?
Show Hint
In geometric probability, the formula is always \( P = \frac{\text{Favorable Area}}{\text{Total Area}} \). Make sure you calculate the correct areas and then simplify the resulting fraction.
Step 1: Understanding the Concept:
This is a geometric probability problem. The probability of hitting a certain region is the ratio of the area of the desired region (the "favorable" area) to the total area of the entire space (the "sample space" area). Step 2: Key Formula or Approach:
1. Calculate the total area of the dart board (a square). Area of square = side\(^2\).
2. Calculate the area of one circular region. Area of circle = \(\pi r^2\).
3. Calculate the total area of the four circular regions.
4. The probability is the ratio: \( P(\text{hit circle}) = \frac{\text{Total area of circles}}{\text{Area of square}} \). Step 3: Detailed Explanation:
1. Total Area (Square):
From the diagram, the side length of the square dart board is 24 inches.
\[ \text{Area}_{\text{square}} = (24 \text{ in})^2 = 576 \text{ in}^2 \]
2. Area of one Circle:
The radius (\(r\)) of each circular region is 3 inches.
\[ \text{Area}_{\text{one circle}} = \pi r^2 = \pi (3 \text{ in})^2 = 9\pi \text{ in}^2 \]
3. Total Favorable Area (Four Circles):
There are four identical circular regions.
\[ \text{Area}_{\text{four circles}} = 4 \times \text{Area}_{\text{one circle}} = 4 \times 9\pi = 36\pi \text{ in}^2 \]
4. Calculate the Probability:
\[ P(\text{hit circle}) = \frac{\text{Area}_{\text{four circles}}}{\text{Area}_{\text{square}}} = \frac{36\pi}{576} \]
Now, we simplify the fraction \( \frac{36}{576} \).
We can divide both by common factors. Both are divisible by 36.
\( 576 \div 36 \): \( 576 = 10 \times 36 + 216 \). \( 216 = 6 \times 36 \). So \( 576 = 16 \times 36 \).
\[ \frac{36}{576} = \frac{1 \times 36}{16 \times 36} = \frac{1}{16} \]
Therefore, the probability is:
\[ P(\text{hit circle}) = \frac{\pi}{16} \]
Step 4: Final Answer:
The probability of hitting one of the circular regions is \( \frac{\pi}{16} \).
