Probability Calculation Using Area Models

Area Calculations

Circle Area = $16 \pi \text{ mi}^2$
Triangle Area = $20(12)/2 \text{ mi}^2$

Probability Based on Area

The probability P of an event occurring based on area is calculated by dividing the area of the specific region of interest by the total area of the region.
- $P = \frac{\text{The region of interest area}}{\text{The total area of region}}$
P is a number between 0 and 1.

Question

What is the probability that a point picked randomly on the figure will be in the shaded area?

Plan

Find the separate areas.
Subtract the circle area from the triangle area to get the shaded area.
Divide the shaded area by the total area (the triangle area).

Calculation

$P = \frac{\frac{1}{2} (20 \times 12) - \pi 4^2}{\frac{1}{2} (20 \times 12)}$
$P = 0.58 = 58\%$

Example 1

A square with a smaller, shaded square.
Probability of choosing a point in the shaded area:
- $P = \frac{P<em>I}{P</em>T} = \frac{s}{x}$
- Given:
  - s = 10
  - x = 3

Solution to Example 1

$P = \frac{9}{100} = 0.09 = 9\%$
Move the decimal point two places to the right to make a percent.

Hexagon and Square

Hexagon Area = $80 \text{ mi}^2$
Square Area = $6 \text{ mi}^2$
P(x) = ?

Solution

$P(x) = \frac{6}{80} = 0.075 = 7.5\%$
Move the decimal point two places to the right to make a percent.

Triangle and Area X

Triangle Area = $25 \text{ mi}^2$
X Area = $15 \text{ mi}^2$
P(X) = ?

Solution

$P(X) = \frac{15}{25} = 0.6 = 60\%$

Areas A and B within a Decagon

Area of A = $10 \text{ mi}^2$
Area of B = $5 \text{ mi}^2$
Decagon area = $100 \text{ mi}^2$

Combined Probability

P(A) means “the probability of choosing a point in A”.
P(B) means “the probability of choosing a point in B”.

Calculations

Calculate P(A).
Calculate P(B).
Calculate P(A or B).
Calculate P(A then B).
- Since A and B are independent, $P(A \text{ and } B) = P(A) \times P(B)$ .

Solutions

P(A) = $\frac{10}{100} = 0.1$
P(B) = $\frac{5}{100} = 0.05$
P(A or B) = $0.1 + 0.05 = 0.15$
P(A then B) = $0.1 \times 0.05 = 0.005$