Chapter 13.3, Problem 39HP

To determine

To explain:

The probability of a randomly chosen point falling in the shaded region of the given two squares is the same.

Answer to Problem 39HP

The probability of a randomly chosen point falling in the shaded region of the given two squares is the same.

Explanation of Solution

Given information:

  

Calculation:

Let X represent a randomly chosen point falling within the shaded region of the square with the inner shaded region.

The probability of a randomly chosen point falling in the inner shaded region of the first square is equal to the area of the shaded region divided by the total area of the square.

  $\begin{array}{l}P\left(X\right)=\frac{ShadedArea}{TotalArea}\\ P\left(X\right)=\frac{0.4375in{.}^2}{\left(1in.\right)\left(1in.\right)}\\ P\left(X\right)=0.4375\end{array}$

Now,

Let X represent a randomly chosen point falling within the shaded region of the square with the outer shaded region.

The probability of a randomly chosen point falling in the inner shaded region of the first square is equal to the area of the shaded region divided by the total area of the square.

  $\begin{array}{l}P\left(X\right)=\frac{ShadedArea}{TotalArea}\\ P\left(X\right)=\frac{1in{.}^2-\left(0.75in.\right)\left(0.75in.\right)}{1in{.}^2}\\ P\left(X\right)=0.4375\end{array}$

Therefore, since the probability that a randomly chosen point falls within the shaded region of each square is $0.4375$ , thus, the probabilities are the same.