Problem 145 - An Introduction to Mathematical Statistics and Its Applications

Question

A point is chosen at random from the interior of a right triangle with base \(b\) and height \(h\). What is the probability that the \(y\) value is between 0 and \(h / 2 ?\)

Step-by-Step Solution

Answer

To get the probability, it is calculated by formula \( P = \frac{A_{\text{lower}}}{A_{\text{total}}} = \frac{1/4 * b * h }{1/2 * b * h} = 0.5\). Thus, the probability that the \(y\) value is between 0 and \(h / 2\) of a point picked randomly from the triangle is \(0.5\) or \(50\%\).

Step 1: Calculate the Total Area of the triangle

We know that, the formula to find the area of a triangle is given by \(A_{\text{total}} = 1/2 * b * h\)

Step 2: Calculate the Lower Half Area of the triangle

Then, we need to calculate the lower half area of the triangle using the same formula, except that the height becomes \(h / 2 , A_{\text{lower}} = 1/2 * b * (h/2)= 1/4 * b * h \)

Step 3: Calculate the Probability

Finally, find the probability, which is the ratio of the lower half area to the total area, \( P = \frac{A_{\text{lower}}}{A_{\text{total}}} \)

Key Concepts

Area of Triangle

The area of a triangle is crucial in geometric probability. It helps us determine likelihoods in exercises like placing points within specific boundaries. For any triangle, the area can be calculated using the formula:

• Area = \( \frac{1}{2} \times \text{Base} \times \text{Height} \)

This formula calculates the space the triangle occupies within its base and height. When dealing with a right triangle, imagine the base as the bottom and the height as the vertical side above it. To fully engage with problems involving area, you need a solid grasp of these basic properties.
The intuition here is straightforward. Imagine slicing the triangle so each piece sits perfectly within the height and base. This methodical breakdown matches our formula, showing how base times height, divided by two, represents the full area. Calculating the area of sections, like halves or quarters, simply involves altering the height or base proportionally.

Geometric Probability

Geometric probability involves calculating likelihoods based on areas or lengths rather than numerical outcomes. For instance, if we have a triangle, geometric probability allows us to find the probability of selecting a point within a specific section of that triangle.
The essence of geometric probability lies in comparing areas or lengths. By taking the area of interest and dividing it by the total area, you derive the probability. Consider the exercise's goal: the probability the point falls within a smaller section defined by its height. This is calculated by:

• Probability = \( \frac{\text{Area of small section}}{\text{Total area of triangle}} \)

In this context, smaller sections might represent significant events or criteria to be studied. For geometric probability, understanding the manipulation and comparison of these areas forms the foundation of probability calculations.

Random Point Selection

When selecting a point at random within a geometric shape, every point should have an equal chance of being chosen. This idea, known as uniform selection, is essential for understanding the nature of random point selection in geometric probability.
To visualize this, picture a right triangle. If you pick a point "randomly" within it, any spot from the base to the peak could equally be chosen.
This equality in chance is managed by considering the area. The probability then spans not only wide horizontal sections but also vertical ones. For example, if asking how likely a point lands in the lower half of a triangle, you're focusing on both y-axis proximity (how high up is the point?) and proportional distribution (how big is that area?).

• Ensure the shape's entire area is accounted for in probabilities.
• Consider smaller divisions to form possible outcomes.

Being able to visualize the entire shape while considering subsets ensures all potential placements are evenly covered, leading to more accurate probability calculations.