Geometric Probability Notes

Geometric Probability

• Geometric probability is similar to regular probability: $\text{Probability} = \frac{\text{Area of what we want}}{\text{Total possible area}}$
• There will be two geometric probability questions on the assessment.
• Show all work, even for simple calculations, to potentially earn partial credit.

Showing Work Examples

• Rectangle: If a rectangle is 8 by 10, write down $A = 80$ around the rectangle.
• Circle: If a circle has a radius of 2, write down $A = 4\pi$. Even if obvious, this shows the area has been calculated.

Homework

• The instructor does not recommend doing the geometric probability homework problems from 9.4.
• Instead, focus on the problems from the review (written by the instructor) and the quizzes.

Using Angle Measures to Find Geometric Probability

• Circles total 360 degrees.
• When using angle measures: $\text{Probability} = \frac{\text{Desired angle measure}}{\text{360 degrees}}$ as total area.

Examples

• Spinner with different colored sections.
• Red (80 degrees): Probability $= \frac{80}{360} \approx 0.22$ (22%).
• Purple (75 degrees) or Blue (60 degrees): Probability $= \frac{75 + 60}{360} = \frac{135}{360} = 0.375$ (37.5% or approximately 0.38).
• Not landing on Yellow (100 degrees): Probability $= \frac{360 - 100}{360} = \frac{260}{360} \approx 0.72$.
• Assessment questions will be more like the example in the bottom half of the slide (using areas of shapes within a rectangle).

Example Problem: Shapes in a Rectangle

• Find the probability that a point chosen randomly inside a rectangle lands within a given shape.
• Rectangle dimensions: 45 x 20.
• Shapes inside: triangle, circle, and trapezoid.

Step 1: Find Areas

• Rectangle Area: $A = 45 \times 20 = 900$. Write $A = 900$ in the corner.
• Triangle Area: $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} bh = 40$.
• Circle Area: radius $r = 6$, $A = \pi r^2 = 36\pi \approx 113.1$.
• Trapezoid Area: $A = \frac{1}{2} \times \text{height} \times (\text{base1} + \text{base2})$. The height is 10, and the sum of the bases is 20, thus: $A = \frac{1}{2} \times 10 \times 20 = 100$.

Step 2: Calculate Probabilities

• Triangle: $\frac{40}{900} \approx 0.04$ (4%)
• Circle: $\frac{36\pi}{900} \approx 0.13$
• Trapezoid: $\frac{100}{900} = \frac{1}{9} \approx 0.11$.

Additional Question: Probability of Landing in the White Area

• Calculate the white area by subtracting the areas of all the shapes (triangle, circle, trapezoid) from the total area of the rectangle.
• Area of white region: $900 - 40 - 100 - 36\pi$.
• Probability of landing in the white area: $\frac{900 - 40 - 100 - 36\pi}{900} \approx 0.72$.

Alternative Method

• Add the probabilities of landing in each of the colored shapes: $0.04 + 0.13 + 0.11 = 0.28$.

• Subtract this sum from 1: $1 - 0.28 = 0.72$. The probability of landing in the white area is 0.72.

• The sum of probabilities for all regions (white, yellow, green, blue) should equal 1 or 100%.

• The assessment may not provide individual probabilities, so calculating the white area directly might be necessary. It may not be set up in a way to do it the easier way, as the assessment itself might not ask for the individual probabilities so you won't already have them calculated.