Geometric Probability Riddle Worksheet - Free Printable

Here are the step-by-step solutions for the problems shown in your worksheet.

Part 1: Geometric Probability (Circle Model)

Goal: Find the probability that a point lands in the shaded region.
Formula: $\text{Probability} = \frac{\text{Area of Shaded Region}}{\text{Total Area}}$

1)
* Step 1: Find Total Area. The shape is a circle with a radius ($r$) of $24\text{ ft}$.
$$A = \pi r^2 = \pi(24)^2 = 576\pi$$
* Step 2: Find Shaded Area. The shaded part is a sector with an angle of $80^\circ$.
$$\text{Area} = \frac{80}{360} \times 576\pi = \frac{2}{9} \times 576\pi = 128\pi$$
* Step 3: Calculate Probability.
$$P = \frac{128\pi}{576\pi} = \frac{128}{576}$$
Divide top and bottom by 64 to simplify:
$$P = \frac{2}{9}$$

2)
* Step 1: Find Total Area. The shape is a trapezoid with height $10\text{ cm}$, base 1 ($17\text{ cm}$), and base 2 ($11\text{ cm}$).
$$A = \frac{1}{2}(b_1 + b_2)h = \frac{1}{2}(17 + 11)(10) = \frac{1}{2}(28)(10) = 140\text{ cm}^2$$
* Step 2: Find Shaded Area. The shaded part is a triangle inside the trapezoid with base $11\text{ cm}$ and height $10\text{ cm}$.
$$A = \frac{1}{2}bh = \frac{1}{2}(11)(10) = 55\text{ cm}^2$$
* Step 3: Calculate Probability.
$$P = \frac{55}{140}$$
Divide top and bottom by 5 to simplify:
$$P = \frac{11}{28}$$

3)
* Step 1: Find Total Area. The shape is a rectangle ($11\text{ mm} \times 4.5\text{ mm}$).
$$A = 11 \times 4.5 = 49.5\text{ mm}^2$$
* Step 2: Find Shaded Area. The shaded part is a triangle with base $11\text{ mm}$ and height $4.5\text{ mm}$.
$$A = \frac{1}{2}(11)(4.5) = 24.75\text{ mm}^2$$
* Step 3: Calculate Probability.
$$P = \frac{24.75}{49.5} = 0.5 \text{ or } \frac{1}{2}$$

4)
* Step 1: Find Total Area. The shape is a pentagon made of 5 triangles meeting at the center. Side length is $10.2\text{ ft}$ and apothem (height of small triangle) is $7\text{ ft}$.
$$A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2}(5 \times 10.2)(7) = \frac{1}{2}(51)(7) = 178.5\text{ ft}^2$$
* Step 2: Find Shaded Area. The shaded part is 2 of those 5 triangles.
$$A = 2 \times (\frac{1}{2} \times 10.2 \times 7) = 71.4\text{ ft}^2$$
* Step 3: Calculate Probability.
$$P = \frac{71.4}{178.5} = 0.4 \text{ or } \frac{2}{5}$$

5)
* Step 1: Find Total Area. The shape is a triangle with base $8\text{ m}$ and height $7\text{ m}$.
$$A = \frac{1}{2}(8)(7) = 28\text{ m}^2$$
* Step 2: Find Shaded Area. The shaded part is a smaller triangle with base $4\text{ m}$ and height $7\text{ m}$.
$$A = \frac{1}{2}(4)(7) = 14\text{ m}^2$$
* Step 3: Calculate Probability.
$$P = \frac{14}{28} = \frac{1}{2}$$

6)
* Step 1: Find Total Area. The shape is a rhombus with diagonals $12\text{ m}$ and $25\text{ m}$.
$$A = \frac{1}{2}(d_1)(d_2) = \frac{1}{2}(12)(25) = 150\text{ m}^2$$
* Step 2: Find Shaded Area. The shaded part is half of the rhombus (a triangle formed by one diagonal).
$$A = \frac{1}{2}(150) = 75\text{ m}^2$$
* Step 3: Calculate Probability.
$$P = \frac{75}{150} = \frac{1}{2}$$

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Part 2: Dart Board Probabilities

Goal: Find the probability of hitting the shaded region on these dart boards.

7)
* Step 1: Find Total Area (Rectangle).
$$A = 28 \times 11 = 308\text{ ft}^2$$
* Step 2: Find Shaded Area (Triangle).
$$A = \frac{1}{2}(19)(11) = 104.5\text{ ft}^2$$
* Step 3: Calculate Probability.
$$P = \frac{104.5}{308}$$
To simplify, multiply by 2/2: $\frac{209}{616}$. Both divide by 11:
$$P = \frac{19}{56}$$

8)
* Step 1: Find Total Area (Large Triangle). Base is $10$, Height is $6$.
$$A = \frac{1}{2}(10)(6) = 30\text{ m}^2$$
* Step 2: Find Shaded Area (Small Triangle). Base is $4$, Height is $6$.
$$A = \frac{1}{2}(4)(6) = 12\text{ m}^2$$
* Step 3: Calculate Probability.
$$P = \frac{12}{30}$$
Divide by 6:
$$P = \frac{2}{5}$$

Final Answer:
1) 2/9
2) 11/28
3) 1/2
4) 2/5
5) 1/2
6) 1/2
7) 19/56
8) 2/5