#### Answer

The probability that a randomly picked American has done 4 years of high school only or is a woman is $\frac{39}{58}$.

#### Work Step by Step

We know that the probability that a randomly picked American has done 4 years of high school only or is a woman:
$\begin{align}
& P\left( \text{completed 4 years of high school} \right)=\frac{(\text{Numbers of students completed 4 years of high school)}}{(\text{Total numbers of students)}} \\
& =\frac{56}{174} \\
& P\left( \text{women} \right)=\frac{(\text{Total numbers of women)}}{(\text{Total numbers of students)}} \\
& =\frac{92}{174}
\end{align}$
$\begin{align}
& P\left( \text{4 years of high school only and a woman} \right)=\frac{(\text{Total numbers women of high school only)}}{(\text{Total numbers of students)}} \\
& =\frac{31}{174}
\end{align}$
$\begin{align}
& P\left( \text{4 years of high school or is a woman} \right)=\left[ P\left( \text{4 years of high school only} \right)+P\left( \text{woman} \right)-P\left( \text{4 years of high school only and a woman} \right) \right] \\
& =\frac{56}{\text{174}}+\frac{92}{\text{174}}-\frac{31}{174} \\
& =\frac{117}{174} \\
& =\frac{39}{58}
\end{align}$
Hence, the probability that a randomly picked American has done 4 years of high school only or is a woman is $\frac{39}{58}$.