A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?
Understanding Probability in Geometric Spaces
This question asks for the probability of a point randomly selected inside a rectangle meeting a certain condition related to its distance from the edges. Probability in geometric problems like this is often calculated as the ratio of a favorable area to the total area.
Calculating the Total Area of the Rectangle
The rectangle measures 6 inches by 5 inches. The total area of the rectangle is simply the product of its length and width.
Total Area = Length $\times$ Width
Total Area = \(6 \text{ inches} \times 5 \text{ inches}\)
Total Area = \(30 \text{ square inches}\)
This total area represents the entire sample space for where the random point can be located.
Determining the Favorable Area: Points at Least One Inch from Edges
We are interested in the probability that the randomly selected point is at least one inch away from any edge of the rectangle. This means the point must be at least one inch from the top edge, at least one inch from the bottom edge, at least one inch from the left edge, and at least one inch from the right edge.
Imagine removing a 1-inch strip from each side of the original rectangle. The points that are at least one inch from all edges must lie within the boundaries of this inner region.
- The original length is 6 inches. Removing 1 inch from the left and 1 inch from the right reduces the length of the inner region by \(1 + 1 = 2\) inches. The new length is \(6 - 2 = 4\) inches.
- The original width is 5 inches. Removing 1 inch from the top and 1 inch from the bottom reduces the width of the inner region by \(1 + 1 = 2\) inches. The new width is \(5 - 2 = 3\) inches.
The favorable region where the point is at least one inch from all edges is a smaller rectangle with dimensions 4 inches by 3 inches.
Favorable Area = Inner Length $\times$ Inner Width
Favorable Area = \(4 \text{ inches} \times 3 \text{ inches}\)
Favorable Area = \(12 \text{ square inches}\)
Calculating the Probability
The probability that the randomly selected point is at least one inch from the edge is the ratio of the favorable area (the inner rectangle) to the total area (the original rectangle).
Probability = \(\frac{\text{Favorable Area}}{\text{Total Area}}\)
Probability = \(\frac{12 \text{ square inches}}{30 \text{ square inches}}\)
Now, we simplify the fraction:
Probability = \(\frac{12}{30}\)
Both 12 and 30 are divisible by 6.
Probability = \(\frac{12 \div 6}{30 \div 6} = \frac{2}{5}\)
| Region | Dimensions | Area (sq inches) |
|---|---|---|
| Total Rectangle | 6" x 5" | 30 |
| Favorable Inner Rectangle (at least 1" from edge) | 4" x 3" | 12 |
Thus, the probability that the randomly selected point is at least one inch from the edge of the rectangle is \(\frac{2}{5}\).
Revision Table: Probability and Geometric Shapes
| Concept | Description | Formula Example (Area) |
|---|---|---|
| Geometric Probability | Probability based on ratios of geometric measures (length, area, volume). | \(P(\text{event}) = \frac{\text{Measure of Favorable Region}}{\text{Measure of Total Sample Space}}\) |
| Sample Space (Total Area) | The entire region where the random point can be located. | For a rectangle: \(L \times W\) |
| Favorable Region (Favorable Area) | The sub-region where the event of interest occurs. | Calculated based on the conditions given (e.g., distance from edges). |
Additional Information: Variations in Geometric Probability
Geometric probability problems can involve various shapes and conditions. Here are a few examples of how the favorable region might be defined:
- Distance from a point: The favorable region might be a circle (or part of a circle) centered at a specific point.
- Distance from a line segment: The favorable region might be a band parallel to the line segment.
- Within a specific shape inside a larger shape: The favorable region is simply the area of the inner shape.
In all cases, the core principle remains the same: calculate the area (or length, or volume) of the total possible space and the area (or length, or volume) of the space that satisfies the condition, then take the ratio.
Remember to carefully define the boundaries of the favorable region based on the problem's constraints, such as "at least one inch from the edge" or "within 2 cm of the center".
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