Answer
The probability that a randomly chosen point in the figure lies in the shaded region in percent form is
[tex]\begin{gathered} p(chosen\text{ }a\text{ }shaded\text{ }region)=25\% \\ \\ \end{gathered}[/tex]
The probability that a randomly chosen point in the figure lies in the shaded region in fraction form is
[tex]\begin{gathered} p(chosen\text{ }a\text{ }shaded\text{ }region)=\frac{1}{4} \\ \\ \end{gathered}[/tex]
Explanation
To find the probability that a randomly chosen point in the figure lies in the shaded region, you need to first find the area of the shape (circle):
[tex]Area\text{ }of\text{ }the\text{ }shape=\pi r^2[/tex]
The radius, r of the shape = 4
[tex]Area\text{ }of\text{ }the\text{ }shape=3.14\times4=50.24\text{ }squared\text{ }unit[/tex]
The next step is to find the area of the shaded region.
The radius of the shaded region = D/2 = 4/2 = 2
[tex]Area\text{ }of\text{ }the\text{ }shaded\text{ }region=\pi r^2=3.14\times2^2=12.56\text{ }square\text{ }unit[/tex]
Therefore, the probability that a randomly chosen point in the figure lies in the shaded region in fraction form is
[tex]\begin{gathered} p(chosen\text{ }a\text{ }shaded\text{ }region)=\frac{Area\text{ }of\text{ }shaded\text{ }region}{Area\text{ }of\text{ }the\text{ }figure}=\frac{12.56}{50.24}=\frac{1}{4} \\ \\ \end{gathered}[/tex]
Also, the probability that a randomly chosen point in the figure lies in the shaded region in percent form is
[tex]\begin{gathered} p(chosen\text{ }a\text{ }shaded\text{ }region)=\frac{12.56}{50.24}\times100\%=\frac{1}{4}\times100\%=25\% \\ \\ \end{gathered}[/tex]
