Definitions: Completed rule \(\displaystyle{P}{\left({A}^{{c}}\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}\) General addition rule for any two events:\(P(A or B) = P(A) + P(B) - P(A and B)\)

Solution

\(Grade\ Most important\begin{array}{l|c|c|c|c} & 4 \mathrm{th} & 5 \mathrm{th} & 6 \mathrm{th} & \text { Total } \\ \hline Grades & 49 & 50 & 69 & 168 \\ Athletic & 24 & 36 & 38 & 98 \\ Popular & 19 & 22 & 28 & 69 \\ \hline Total & 92 & 108 & 135 & 335 \end{array}\)

S = Sixth grader G = Grades We note that 135 of the 335 people in the table are 6th grades, because 135 is mentioned in the row ” Total” and in the column ”6th grade” of the given table. The probability is the number of favorable outcomes divided by the number of possible outcomes: \(P(5th\ grade) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{135}{335}\) We note that 168 of the 335 people in the table rated good grades as important, because 168 is mentioned in the row ” Grades” and in the column ”Total” of the given table. \(P(Athletic\ and\ 5th\ grade) = \frac{\# of\ favorable\ outcomes}{\# of\ possible\ outcomes} = \frac{168}{335}\) We note that 69 of the 335 people in the table are 6th graders who rated good grades as important, because 69 is mentioned in the row ” Grades” and in the column ”6th gradel” of the given table. \(P(G)=\frac{\# \text{of favor about comes}}{\# \text{of possible out comes}}=\frac{69}{335}\) Use the general addition rule: \(\displaystyle{P}{\left({S}{\quad\text{or}\quad}{G}\right)}={P}{\left({S}\right)}+{P}{\left({G}\right)}-{P}{\left({S}{\quad\text{and}\quad}{G}\right)}\)

\(\displaystyle={\frac{{{135}}}{{{335}}}}+{\frac{{{168}}}{{{335}}}}-{\frac{{{69}}}{{{335}}}}\)

\(\displaystyle={\frac{{{135}+{168}-{69}}}{{{335}}}}\)

\(\displaystyle={\frac{{{234}}}{{{335}}}}\)

\(\displaystyle\approx{0.6985}\)

\(\displaystyle={69.85}\%\)