Chapter 9, Problem 14PS

To determine

To find:The probability that the dart hits the region outside the triangle.

Answer to Problem 14PS

The probability is $\frac{\pi -1}{\pi }$ .

Explanation of Solution

Given:

The circular target is shown in the figure below:

  

Figure 1

Calculation:

The probability of happening of an event E is given by the formula,

  $P\left(E\right)=\frac{\text{cases in favor of even }E}{\text{total number of cases}}$

From Figure 1, the radius of the circle is 6, the height of the triangle is 6 and the base if 12, that is, the diameter of the circle.

Now, the total number of cases represents the area of the entire circle, with radius 6. So,

  $\begin{array}{c}\text{total number of cases}=\text{area of the circle}\\ =\pi {\left(6\right)}^2\\ =36\pi \end{array}$

Let the event of the dart hitting the region outside the triangle be represented by E . Then,

  $\begin{array}{c}\text{cases in favor of event }E=\text{area of circle}-\text{area of triangle}\\ =36\pi -\left(\frac{1}{2}\times 12\times 6\right)\\ =36\pi -36\\ =36\left(\pi -1\right)\end{array}$

So, the probability that the dart hits the region outside the triangle is,

  $\begin{array}{c}P\left(E\right)=\frac{\text{cases in favor of even }E}{\text{total number of cases}}\\ =\frac{36\left(\pi -1\right)}{36\pi }\\ =\frac{\pi -1}{\pi }\end{array}$